I've redacted the transcript to just highlight the important bits. The short answer is that neither Amy or River are especially keen on the idea of leaving Rory on his own in the past. Both of them are (rightly) convinced that if she lets the Angel touch her, she'll rejoin her husband.
Although the Doctor is sad about this (as well as worried about her personal safety since he's unsure where/when she'll turn up), he has no intention of physically stopping her from making her own decision.
The Angel standing behind RORY touches him and he disappears.
DOCTOR: I'm sorry, Amelia. I'm so, so sorry.
AMY: No. No, we can just go and get him in the TARDIS. One more paradox.
DOCTOR: Would rip New York apart and I --
With shuddering breaths, AMY walks towards the Angel.
DOCTOR: Amy? What are you doing?
AMY: The Angel, would it send me back to the same time, to him?
DOCTOR: I don't know. Nobody knows.
AMY: (takes a step closer) But it's my best shot, yeah?
AMY: It'll be fine. I know it will. I'll -- I'll be with him, like I should be. Me and Rory, together. Melody. (holds her hand out behind
AMY: (sobs) Raggedy man, (turns to the DOCTOR) goodbye. (disappears)