How do I support my 19 year old Aspie son?

I’m sorry - I can see that you are trying to do the right thing - but it’s not always easy to know what that is in practice. I have two autistic sons in their twenties - and I’m autistic too. Like anyone else their needs change as they get older - so it’s not a straightforward process to always know how best to be supportive. Please don’t be too hard on yourself - you obviously love your son and are doing your best. 
I think a key thing in any relationship is to truly listen (TRULY, deeply LISTEN) to what the person is communicating to you (either verbally or non verbally). Often we think we’re listening (and are genuinely trying to) when in some ways we are not fully acknowledging and truly tuning in to what’s really being communicated to us. Also - the person really needs to know that this is your 100% heartfelt  intention- to truly listen and acknowledge how they feel and respect their experience.

The comment you refer to that your son said about ‘if you don’t understand me who will’ really does sound like a plea to you to connect with him. It’s actually wonderful that he wants to do this - and shows that your relationship with him still means a lot to him. 
If I were in your shoes I would essentially say to your son : “ I know that our relationship hasn’t been that good recently, and I know we’re both struggling in various ways, but can we wipe the slate clean and start again? Put past tensions behind us and have a fresh start? Because I love you and our relationship and your happiness means the world to me, and I want to do all I can to improve things and to be there for you?’. Basically : ‘beginning anew’. 
It’s never too late to start again and rebuild a relationship if the will to do so is there. The important thing is: forgive and forget whatever went before. Make it 100% clear that you’re on his side no matter what. Acknowledge that you’ve made mistakes and drop all judgment on both sides. 
Autistic people are like anyone else in that they want to feel understood, acknowledged and respected. Life can be incredibly hard for teenagers - especially autistic teenagers. Make it clear to him that you understand that, and that you don’t judge him for struggling, or for finding things difficult that other people maybe find easy or obvious. 
Autistic people often feel like total failures because they struggle with so many aspects of life - and they need people in their lives that don’t judge them for that and that instead celebrate who they are no matter what. Unconditional love - that’s what we need from our loved ones. 
I really hope things improve for you both - it’s well worth the effort and time to improve this very precious relationship we that have with our children - and they need us - even as adults we still can benefit from a good relationship with our parents. Good luck! . 

Thank you so much for your kind words. I think you are right, I have got into the habit of not really listening, especially as it always ends in a fight. I probably started expecting so much more from him because he is older and I see that it must be coming across as me judging him, which makes me feel terrible. A reset would be amazing but it will take a lot of courage, what if it doesn’t work and we end up in a worse place. 

Thank you so much for your kind words. I think you are right, I have got into the habit of not really listening, especially as it always ends in a fight. I probably started expecting so much more from him because he is older and I see that it must be coming across as me judging him, which makes me feel terrible. A reset would be amazing but it will take a lot of courage, what if it doesn’t work and we end up in a worse place. 

I don’t think a ‘reset’ is likely to make things worse if it’s done with good will on both sides. Bear in mind that your son is probably feeling very fearful of many aspects of life and most likely is eager to have things much better between you. Arguments are stressful and it’s the last thing either of you want I’m sure. Apologise to him - tell him you’re sorry if you’ve been putting pressure on him and making him feel worse - when in truth you’ve just felt confused about how to be supportive and do the right thing. I think the thing to do is to rebuild trust - trust is so important in all relationships. Admit that you’ve made mistakes. Maybe share with him fears that you have in your own life. Don’t be afraid to show your own vulnerability. You really can improve this relationship I’m sure - your son clearly feels a wish to have you on his side and to understand him - as is demonstrated by the comment he made that you mentioned in your post. He needs you. Be there for him unconditionally. You really can do this - you say it would ‘take a lot of courage’ - but what do you have to lose? Being really nice to your own child involves very little risk surely?! Focus on compassion, forgiveness and kindness.