I'm giving my own view of the matter which is the answer that Ayn Rand might have given.
I will have to reformulate your questions, as otherwise the "without objection from anyone" will make any attempt moot: You cannot prove even the simplest and most obvious things in a way so that everyone agrees with you.
But agreement isn't the standard of objectivity. Just take modern science. I'd say that having landed on the moon and created the internet with sites such as this one makes a case that at least some of the sciences have objective truth, as they are used extensively in the technology that brought these things about.
That doesn't mean that no person on the planet still holds superstitious beliefs. In fact I would even claim that a great many people exhibit a strange attraction to the irrational rather than a curiosity about the unknown.
The question whether morality is objective has to be asked under the same light: Not if you can convince a majority (or in fact anybody) of the objective correctness of certain moral values, but whether they are of use to you - in other words, whether they are practical.
The objective standard of correctness for your science of engineering is whether you can built cars with it and airplanes - the objective standard of your science of morality is whether you are guided by it through your life on a path that makes your life good.
The question of what a good life is is much harder, but in this context it suffices to show that there is something of an objective standard: To be sick, unhappy, and to die is bad, to make pleasurable experiences, to be healthy and happy is good. This goes for animals and humans alike and those things are objectively true - I doubt that anybody disagrees here.
The difference to non-human animals is only that human beings have more reason, plan more ahead and thus need rational guiding principles to lead a good life. For example, if someone simply doesn't understand the concept of ownership and just takes what he pleases, he will find his victims to organize against him. This is independent of a specific legal framework and holds in modern societies as much as in the most primitive of tribes. In fact the legal frameworks in modern societies are this organized defense against (objective) immorality to a great extent.
It's more difficult to give a positive example, but say someone was a curious child, learned as an adolescent, became productive, made rich life experiences, founded a company and a family and eventually died of old age among his loved ones. That's better, and fairly objectively so. If someone begs to differ even here on this last example being better than the former, he shall prove so by starting to steal.
There are two more interesting concerns generally raised at this point. The first is that it's not clear what exactly constitutes "better" in many cases. Let's not be distracted by this question: It's also not clear how one would order "mistakes" in natural sciences. That doesn't mean that there is no right and wrong, no more or less fitting models of the universe or the weather. It is completely sufficient to understand that death is bad and life is good - objectively so - and that all more complex questions have an answer in between, no matter how difficult the answer might be.
The second concern is regarding the clash of this view of ethics with the conventional one in cases where they contradict. For example, one often hears: "What if I can steal without getting caught? Is it moral then?"
Yes. But you can't. At least not for long.
Everyone needs other people, but career criminals need other people as victims: That means they need someone who doesn't benefit from this "relationship". Maybe the criminal can overpower or outwit his victims' resistance for a while, but ask yourself honestly: Given the game theoretical setup here, do you think becoming a career criminal is an intelligent career choice? At least it's obvious that most people don't choose that path. Many might claim altruistic motives for such a decision and I concede they might themselves even believe that. I would still hold that the fear that such a decision probably is a really bad idea helps them in their choice. Such a selfish motivation just doesn't sound very heroic - in fact it sounds rather cowardly - which could be one reason why people often really don't like that kind of reasoning.
There are a lot more objections along that line that would justify being answered, but I don't want to make this answer excessively long. I would maintain that I said enough to make a point for objective morality. The most important thing to realize is that morality isn't just "rules we can all agree on" (although that can be a part of it). In doesn't start with other people at all. It's how you, the researcher of ethics, can improve your life - and getting along well with others will obviously be only a part of that, just as defending yourself from certain other individuals is too.