You did absolutely nothing wrong by looking him up and I'm surprised he was annoyed. Ts are so naive about this stuff. Still it's great you told him and hopefully he can help you through this.

Try to think about the reality for most new parents who have careers a little bit. You might be surprised but it may be reassuring. The fact is that as a male it's highly unlikely he will take months off. It's just not the culture in the US for fathers to do this - mostly because mom will need it more than him and someone needs to get paid. Exceptions aside, many employers don't offer paid paternity leave. Even paid maternity leave is not something employers are legally required to provide. Most of the time new moms are on ST disability leave which is 6 weeks. After that, it's vacation and then it's all unpaid. So while your T will undoubtably take some vacation time off, I'm confident that it won't be too much. I'll bet that's why he was so adamant that he won't be leaving you - with a new baby he will probably work more, not less. Especially if he's in private practice. So while you'd think it might take him away from you it's just the reality for most working people.

As for his suggestion that you see a pdoc, whether to go or not is totally up to you. Your T was merely suggesting it because he is clearly concerned. Your anxiety level is very high so maybe your T thinks meds might be helpful. Regardless of his intentions and what you decide to do, a pdoc would only be to supplement to your therapy, not a replacement.