On usual incognito rounds of His kingdom, he hears people talking about Sita. They doubt that any woman, even if it were Sita, would not have succumbed to a man, especially after she was under his control for months together.
(image courtest nirmukta.com)
A lot of issues at hand here, some political:
Firstly, I don't think that Sita should be viewed as the sort of helpless woman she is portrayed as normally. "She crossed the lakshman rekha herself", "she wanted a deer that attracted her", etc.
Focusing on the agni pariksha incident itself, I don't know if Valmiki goes into details on the Rama-Sita conversation after Rama learns about his citizens' doubts. I have a strong feeling that Sita herself suggested to Rama that the best course for him as the king, and her as queen, is to subject herself to agni pariksha. While the husband may have no doubts on his wife, one holding public office does need to take his citizens' views into account.
(image courtesy awara32.blogspot.com)
That said, I can't understand why Sita had to leave Rama. Having stayed with him through hardships a princess would not be accustomed to, being abducted, and so on, why did she have to leave him after the agni pariksha? Somehow a perfect marriage and couple seem slightly imperfect. She didn't seem to question any of his actions until then - not why he came late to reclaim her, not why they had to go to the forest in the first place. Why question his decision all of a sudden?
Curiously, I don't know that Janaka (the father of Sita) expressed any opinions on his daughter's situation - to Rama or in general. Not sure if Valmiki goes into the details, but I'd be interested in learning more. In today's world, any of what Sita had to undergo would be what the father-in-law might use as grounds to push for a divorce. Was Janaka silent because he knew that his son-in-law could do no wrong? Surely, seeing one's daughter undergo an agni pariksha in public to prove her chastity must have been unbearable? (Was he already dead by that time?)
Finally, Sita does not go back to Janakapuri (which by the way is in present day Nepal). Sita goes to Valmiki's ashrama instead, and ultimately raises her two sons there. Had she gone back to her father, that would have indicated that their marriage had failed which it hadn't. Anyway, overall Sita leaving Rama and moving to Valmiki's ashrama seems to me to have served no major purpose, other than punishing Ayodhya's citizens and depriving them of the presence of their queen and of their twin princes.