There is no way to know how long recovery takes from such news, or how it will change his life.

First of all, I would ask if you believe this news. Getting a paternity test seems straightforward: Is he doing that?

Assuming that you believe him, I would not suggest that you keep doing what you are doing; it does not seem to be working.

I think it is a little early to give up and walk away, under the circumstances, if you are still interested. A third option is just to wait, to give him time and space. You could decide to wait and see where things are in two, three, four weeks. You choose the length of time that you are willing to wait. You could tell him to contact you in the next few weeks if/when he is ready. If he doesn't contact you within the time you set for yourself, then you might decide to give up and move on.