The Red Devils wasted their get-out-of-jail-free card at the end of a wretched domestic campaign as the side 16th in the Premier League lost 1-0 to 17th-placed Spurs at San Mames on Wednesday night.
"What I can say is that I think it was clear we were the better team, but then we managed not to score again, and then it's really hard to win football matches," Amorim said.
"But the guys tried everything to win the game."
United's first Europa League defeat of the season came at the worst possible time and means next term they will be without continental football for the first time since 2014/15.
The meek loss increases the spotlight on Amorim, who has won just six Premier League matches since succeeding Erik ten Hag as manager in November and says he will go if the club do not think he is the right man for the job.
"In this moment I am not going to be here defending myself," the Portuguese declared.
"It's not my style, I cannot do it ... If the board and the fans feel that I am not the right guy, I will go in the next day without any conversation about compensation. But I will not quit.
"I am really confident in my job and, as you can see, I will not change anything in how I do things."
United have been sent into a tailspin by the loss in Bilbao with the lack of Champions League football - worth an estimated £100 million ($A208 million) - damaging their summer rebuild.
"We need to understand that it is tough for a club like ours not to be in the Champions League," Amorim added.
"But ... we have more time to think and to work during the week and to be better in the Premier League. That will be our focus."
Club legend Rio Ferdinand told TNT Sports: "It's been a disaster of a season.
"I thought both teams looked scared to lose it ... you have to be brave in these moments. It's going to be a long, long summer.
"That recruitment list they had will have to be scrapped now because they are not going to have Champions League football. It is a completely different list to recruit from."
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