suggestions are: ParkRun. Saturday mornings in your area. Can be as solitary or as sociable as you want.
There is nothing you can do to force someone to have an assessment, even if he does has ASD it won't be a magic bullet just because you "know". However, in light of your suspicions, you could try and encourage socialising that is more tailored : ie local Warhammer group, science club at school. try and engage in his interests, find articles in paper that might interest him, discuss them with him. Often parents are first step to conversations that you might end up having with peers. This is what happened with ds2 (ASD) rubbish at socialising to start with, slowly built up from 13, now is able to get on with a few friends without boring them to death, and actually they really value his humour and energy. His main interests are football - this has been the conversational glue, despite him being very bad at playing it, he attends every match and even away games now. We had to do a lot of facilitating to get to this point. Same with drama, he doesn't star in any productions, refuses to get involved in school drama except in classroom, but loves seeing funny plays and musicals/pantomimes (hates reading, hates literary discussion). I have had to sit and watch things with him like Mr Selfridge, Play that Went Wrong and dh has sat through a 1000 freezing football matches, but it has helped him socialise and have confidence ultimately that he has enthusiasms that others share.
16 is pretty young - I know other people whose sons blossomed around 21, young people with asperger's can often lag developmentally on the socialising side but eventually get there, jsut don't panic and do small things/ appreciate small gains and do not compare them to peers, who are going to parties, having girlfriends and generally breaking away/becoming super independent. It takes time to get there.