Sunderland's relegation to the Championship will be confirmed if they lose to Bournemouth on Saturday and 17th-placed Hull City get a point at Southampton.
Fans chanted "we want Moyes out" during the Boro game.
"I'm here. I'm the manager. I'll take it on the chin," Moyes told reporters after the match. "There's nobody who wants to win more than me.
"I'm used to winning, I'm not used to losing, and I don't want to get used to it either."
Sunderland fans also chanted "You're not fit to wear the shirt" and booed their players after the final whistle but Moyes said there is still hope of survival.
"While there's a chance, I've got to keep going and we've all got to keep going," the Scot said.
"We are not enjoying it, we are not enjoying losing or the position we are in.
"Criticism is rightly due, but I wouldn't do it on tonight's performance. The players have stuck at it and shown they are capable of trying to get points and winning games."
(Reporting by Aditi Prakash in Bengaluru; editing by Sudipto Ganguly)
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