Responding to requests to turn my comments into an answer
The question of whether anything exists when we are not observing it is a standing question in philosophy. As Philip points out in the comments, this is a topic which nearly every major philosopher has written multiple books about, so an answer which fits within the space of a StackExchange post will be hard to come by. Needless to say, there is not a consensus to the answer.
Part of what makes the topic difficult is that the concept of observation is so intuitive in the simple human-sized cases. I look at the moon; I look away; I look back -- it's still there. However, if you start to explore what it means to observe something you quickly enter the world of epistemology, the study of knowledge. Epistemological approaches typically admit that they are not in a position to make true/false statements about reality. That's the realm of ontology, not epistemology. In fact, as one dives deeper, the question of what an observation is gets even murkier, not clearer.
Fortunately, as someone with a physics background, you have access to a set of experiments where observation is known to have quirky behavior: quantum physics. You can set up many experiments which demonstrate that the intuitive concepts of what an observation is break down. In fact, they break down so thoroughly that QM scientist typically call them "classical observation," to indicate that the concept is really only valid in scenarios where classical physics applies. In their world, they instead play with concepts like expectations and covariances and superpositions. These are statistical constructs which are very effective at predicting results, but often leave one pondering whether there is something more. These experiments may be very helpful in shaking up one's intuitive understanding of observation. This may permit one to explore the philosophical questions regarding observation with a more open mind.
My favorite series of these experiments is the quantum eraser series. I like them because they truly wrack the intuitive mind, while demonstrating numerical results that are very well predicted statistically by quantum physics. However, they leave one pondering how in the world these statistics could actually describe reality, until one leaves behind concepts such as the idea of classical observation. The experiments are quite detailed, but I will dare try to offer the cliff notes:
The double slit experiment - Monochromatic coherent light, such as that from a laser, is passed through a beam spreader onto a surface with two parallel slits in it. These allow the light to pass through onto a second surface some distance behind it. Due to differences in the path length between the distance from the laser to each slit to the spot on the paper, interference patterns show up. Instead of seeing two bands, as one might expect, one sees a series of interference fringes. This demonstrates that light travels in waves. Great. Nothing fancy QM here. Just a proof that light behaves like a wave. We can observe it (classical observation). Moving on.
Single photon double slit experiment - Now lets tune the laser down. Instead of shining a ton of light, let's just shine a little. In fact, we can send just a single photon at a time by providing just enough power to emit a single quantized packet. How do we know it's quantized? We can do experiments regarding the photoelectric effect to demonstrate that we cannot generate a photon outside of its quantized energy region. At the very least, we can show we don't have two photons... just one. Now the photon goes towards the double slit, goes through one slit, and hits the board. Easy peasy. Only one problem. If you do this multiple times, and start looking at the probability of the photon appearing at any given position, the results don't line up with that intuition. Instead, you see a spattering of points which lines up perfectly with the interference patterns from the earlier experiement. Somehow the photon is seeing interference, even though there's no other photon for it to interfere with besides itself. This gets explained as wave/particle duality. The photon acts like a particle and a wave. QM has equations to explain the exact pattern we see, but the intuitive mind may tweak things to accept that there is a thing which can travel through both paths at once. No issues here. This is kinda creepy, but we're not seeing any really unusual observation like behavior. The moon is still there, it just behaves differently than we thought.
Double slit experiment with path detection Now it starts to get creepy. We use a nonlinear optical material to split the photon into an entangled pair. Why this works is hard to explain without QM equations to back it up, but experimental evidence supports the theory that some materials can actually split photons into an entangled pair of photons, each with half the energy. The super long name for this process is spontaneous parametric down-conversion and it occurs in a material such as beta barium borate and I will happily admit that I have no idea how this happens and am comfortable accepting that empirical studies have demonstrated that this works. This is done before the photon hits the double slit. One entangled photon goes to a detector, while the other goes to the double slits. Nothing unusual happens. You still see a double slit interference pattern. The next step is where it starts to get creepy. We put circular polarizers in front of each slit. Thus the polarization of the photon streaking towards the detection grid after the double slit contains information about which path it took. Because this is entangled with the other photon, the other photon must have the same polarization. If we put a circular polarizer in front of the detector of this entangled photon, we will only capture photons with a particular polarization. Now a curious thing emerges. When we only look at photons with a particular polarization (meaning a certain path), the pattern we see on the detection grid ceases to show interference patterns. It shows a straight forward distribution with one big lobe, like one would expect if light behaved like a particle. This is the first time observation starts to act unlike what we would like: somehow the mere fact that we observed the photon's path caused it to cease to interfere with itself!
The quantum eraser - Now do the same experiment, but put a linear polarizer in front of the detector of the entangled photon. This destroys any path information we might have had, because clockwise and counterclockwise polarized light have equal probabilities of passing through a linear polarizer. Suddenly, the interference pattern jumps out again. The photon knew that it's path wasn't being observed, so it went through both slits again. The previous sentence should raise hives on the skin of any scientist. Photons can't know things. Somehow this "observation" thing is doing far more than it really should. Once again, the QM equations predict these reults with statistical perfection, but its hard to explain how any one result could occur. The statistics make sense, but the individual cases are baffling.
Delayed quantum eraser - What if we extend the path length to the entangled photon detector? What if the entangled photon hasn't even reached the linear polarizer before it gets detected. Bafflingly, it doesn't matter. We still see the interference pattern emerge. Remove the polarizer, the interference patterns vanish. Now it's getting creepy. Now, not only do photons appear to know whether they have been observed, they seem to know whether they are going to be observed. Classical observation falls flat on its face here. The only way to explain what we see is through quantum mechanics, intentionally avoiding "observation" of anything except for the final product. There's nothing magic here, just the concept of observation ceasing to be effective.
Delayed choice quantum eraser - God save whoever thought this was a good idea (Yoon-Ho Kim, R. Yu, S.P. Kulik, Y.H. Shih and Marlan O. Scully, to be specific). They swapped up the delayed quantum eraser, putting the double slit first, then using beta barium borate to split it into entangled photons after each slit. One entangled photon went to the detection grid to be measured. The other, which they called the "idler photon" went into a mousestrap of beam splitters and detectors. 3 beamsplitters each reflected the photon 50% of the time, and transmitted it 50% of the time. They organized them very carefully to capture different path information depending in which detector it hit.
- The only way to hit one detector is for it to go through one slit
- The only way to hit another detector is for it to go through the other slit
- The two remaining detectors could be triggered by a photon through either slit.
What's different here? When the photon hits the detection grid, it is literally unknown what detector its entangled sister will go to. There's a 50% chance that it will go to a detector which reveals which path it took. There's a 50% chance that it goes to a detector that does not reveal anything, "erasing path information."
The result? If they filter out only the photons whose sister hit a detector which gave path information, the result was a wide pattern with no interference patterns -- particle behavior. If they filter out only the photons whose sister hit a detector which erased path information, we saw interference patterns.
If one thinks in terms of classical observation, somehow the photon must have "known" which detector its entangled sister would go to in the future! It would appear that a future event affected the past detections! Now not only are observations affecting the result, but observations that we don't even know if we're going to make seem to be affecting it! Once again, the QM equations predict the results without any of this future observation nonsense, but they have to do so statistically, by using terms like superposition of states. Trying to explain what any one photon will do is fraught with error!
After exploring the exotic results that occur with the series of eraser experiments, hopefully that helps unseat any privileged status that "observation" might have. After that, accepting that philosophy finds it a tricky topic may be more palatable. Then you could explore what individual philosophers thought "observation" or "perception" might be, and the consequences of those beliefs. That's where the real fun begins!