This is awful. So much of the time the Dad gets the bad end of the deal. You sound like you really love your son - you don't sound like a 'Jeremy Kyle regular', so it sounds completely unfair to me.
However, there are always two sides to every story. It seems odd that she's flown off the handle just because of one incident. Are there other times when you haven't been able to look after him, or it's come across like you don't want to look after him? I'm not saying this is the case, I'm just wondering if you've tried to think of anything you could have done other than this to make her take away access to your child? It will be really important to think about all of this if you do decide to see a solicitor, because EVERYTHING will be used against you.
I worked as a legal secretary in the past and did some work for a solicitor who deals with family issues. There was a case similar to yours, where the mother had stopped allowing the father of the child to see their daughter because she said that the father only wanted to see the daughter when it suited him, and this was disrupting the child's life. She even said that he was an unfit father because he drove a high-powered sports car, which she didn't believe was safe for their daughter to be driven around in. See what I mean when I say EVERYTHING is taken into account?
Whatever's happened in the past (if anything has happened at all), you have every right to see your son. I really feel sorry for you, as you obviously miss him.
I don't know where you live, otherwise I would have a look for some law firms in your area for you - but just google 'solicitor' and have a look through. It will probably be an expensive thing to do, but it means you'll have legal rights to your son - right there in black and white. Although, you should try to make every effort to keep things civil with your ex and avoid things getting bitter - for the sake of your son. Although she does come across as rather spiteful, so I feel for you. You should also get a DNA test done if this is possible, as it won't matter whether your name is on the birth certificate if you have proof that he's your son. Incidentally, why isn't your name on the birth certificate? Even if you can't prove he's your son, there have been cases where non-relations or grandparents, etc have gained custody of the child instead of the mother, so you DO have a leg to stand on, I'm sure. All you have to do is prove to them how much you love him.
In the meantime, maybe you could try writing your ex a clear, calm letter. If you have made mistakes in the past regarding your son, apologise for them. Explain that it really was an honest mistake. Point out all the times you've done her a favour by having him outside of your regular times and things like that. Tell her how much you love and miss your boy. You don't have to grovel to this woman, but appeal to her better nature and just give it a shot before taking legal action. And whatever you do, don't fall into the trap of talking your ex down in front of your boy, or trying to turn him against her. Even if she starts doing this, which it sounds to me like she would do (she sounds like the 'type'). Don't let it phase you. Don't drop to her level. If you hear anything of this sort from your son as he grows up, just tell him that you love him and that mommy made a mistake. He will grow up learning that you are the fairer parent.
If all else fails, yes, get yourself a lawyer. You have every right to see your son.
Good luck and I hope this helps.