Some searching for studies on Google Scholar will turn up any number of articles which suggest humans have evolved as a social species with cognitive faculties which make them better adapted to interact with other humans, and thrive in environments with positive social interaction.
For instance, Herrmann, 2007
we found that the children and chimpanzees had very similar cognitive skills for dealing with the physical world but that the children had more sophisticated cognitive skills than either of the ape species for dealing with the social world.
This suggests that our mental abilities imply that we live in complex social communities, and are oriented to survive in those communities.
A history of emotional abuse and neglect was associated with increased anxiety, depression, posttraumatic stress and physical symptoms, as well as lifetime trauma exposure. Physical and sexual abuse and lifetime trauma were also significant predictors of physical and psychological symptoms. Hierarchical multiple regressions demonstrated that emotional abuse and neglect predicted symptomatology in these women even when controlling for other types of abuse and lifetime trauma exposure.
This is a bit more of a stretch scientifically because it's not isolated to neglect, but neglect is the key word here. Children who are socially neglected experience, on average, more mental disorders over the course of their lifetime.
It also suggests that negative social interaction is associated with psychiatric problems.
This paper explores the role of domestic technologies for addressing social isolation of older people. Despite the increasing use of information and communication technologies, social isolation remains an issue amongst older people. Assistive technologies address important health needs, but there is a lack of social technologies that adequately deal with social isolation. This paper contributes to knowledge about the everyday life of older people for the purpose of designing appropriate social technology.
This paper implies that social isolation is a problem with senior citizens, and investigates ways to deal with that issue. You could go further and assume from this that, neurologically, isolation is a problem.
These are just a few examples, but I'm sure I could go on. Maybe there is something else that can be said about this, but science does say that if you're socially isolated you're going to run into problems.
So to directly answer the question 'can people live happily alone on a deserted island', I'd suggest that the answer is no, they can't. This line of thinking over-estimates our ability to and runs right in line with our cultural belief that people can think their way out of their problems.
Maybe people can think their way to happiness to a certain extent, but generally the most effective way to resolve problems is to actually get rid of the problem itself. And the studies above definitely suggest that social isolation and abuse are serious problems.