Travelling supporters sang 'We want Moyesy out' at the Riverside Stadium as Marten de Roon's early goal left the Black Cats 12 points adrift of Premier League safety and staring relegation full in the face with only five games remaining.
But asked if he would go, Moyes, who earlier in the day was hit with a Football Association charge for comments he made to a female reporter, said: "No, I'm here, I'm the manager, you take it on the chin.
"There is nobody who wants to win more than me. I am used to winning, I'm not used to losing and I don't want to get used to it either.
"I'm a football supporter, I know what it's like. You don't like seeing your team lose."
Asked if his job had been made more difficult by Hull's resurgence under new manager Marco Silva, Moyes added: "It's good when people come in, but when people do it for 14, 15 years in the Premier League, and how their record stands up over that period, that's when you judge people."
Sunderland remain 12 points shy of 17th-placed Hull and must at least match the Tigers' result at Southampton on Saturday when they face Bournemouth at the Stadium of Light if they are to keep their weak survival bid alive.