United manager Solskjaer is facing even more frenzied speculation over his future after a fifth defeat in seven Premier League games.
The club's hierarchy stuck by Solskjaer during the recent international break despite humiliating home defeats to Liverpool and Manchester City.
But worse was to come at Vicarage Road as the struggling Hornets could afford to miss a first-half penalty twice and still run out comfortable winners.
"I feel for the fans and I feel the same as them. We're embarrassed by losing the way we do," Solskjaer told 'Sky Sports'.
"We know we are in a very bad run and a bad situation, but that's football and I know they'll support whoever is on the pitch every day, and sometimes you've got to say sorry."
While United's owners, the Glazer family, have stayed loyal to Solskjaer so far, they may feel compelled to wield the axe after the latest humiliation for a team spiralling into crisis.
Solskjaer insisted he was not concerned about speculation over his future, but he did concede United's current results were unacceptable.
"I'm working for and with the club and I've done that for 18 years," he said.
"We've got a good communication and if the club were thinking about doing something then that is between me and the club.
"The results are not good enough, we know that. We've gone 30 games unbeaten away from home and now we've lost two on the bounce and conceded four goals in both of them, so of course something's wrong."
'Hard to watch'
United goalkeeper David De Gea was even more scathing in his assessment of the timid display.
"It was embarrassing to see Man United play like we did today," said De Gea, who saved two spot-kick attempts from Ismaila Sarr after the first was retaken.
"It's not acceptable - the way we were playing and doing things. It's easy to blame the manager or the staff but sometimes it's the players. We need to show much more than we are doing.
"Embarrassing first half - we could've conceded four goals in 45 minutes. It was hard to watch the team playing today - it was nightmare after nightmare. It's not acceptable."
Joshua King and Sarr gave Watford a 2-0 half-time lead before Donny van de Beek came off the bench to halve United's deficit.
However, any hope of a fightback was thwarted by a red card for captain Harry Maguire and late goals from Joao Pedro and Emmanuel Dennis rounded off another humbling afternoon for Solskjaer's men.
"It's everyone's fault," said United midfielder Bruno Fernandes. "It's not the coach, it's not one player - it's everyone."
Solskjaer, though, is likely to be the fall guy with a Champions League trip to Villarreal to come on Tuesday before daunting Premier League clashes against leaders Chelsea and an in-form Arsenal.
"This is not Manchester United or the way we ought to play - it's difficult to say more than that," added De Gea.
"It's been very bad for a long time - a club like Man United we have to be fighting for trophies and fighting for big things and to be honest we are far from that.
"We always say the same things but it's the truth - we have to look at ourselves and see where we can improve."
Standout matches today