Villa held off a strong Tottenham fightback in the second half after goals by Emiliano Buendia and Morgan Rogers had put them in control by halftime in north London.
Wilson Odobert halved the deficit with a low shot after 54 minutes and Tottenham showed the fighting spirit that was sadly lacking in the first half, although it did spill over.
Tottenham midfielder Joao Palhinha was enraged as Watkins and other Villa players went to celebrate in front of their fans, with players from both sides involved in ugly scenes.
"I saw what I saw. Of course it's all about keeping a cool head. The players gave everything, everything out there, (but) losing a tight game, season not going perfect and I think Ollie (Watkins) is very provoking," Frank told reporters.
"The way he is going down to celebrate in front of the Villa fans and he is walking into Joao, he can just easily walk around. I think everyone that has been in a competitive nature, that is difficult and can trigger things."
The FA Cup exit deepened Frank's problems as his side again failed at home, having won only twice in 10 Premier League games in front of their own fans.
After the 3-2 loss at Bournemouth in midweek several players became involved in heated discussion with fans while captain Cristian Romero was suspended on Saturday after getting an extra one-match ban for acting improperly by failing to leave the field of play promptly after his red card against Liverpool.
Asked if his side have a discipline problem, Frank said: "When you play with passion, then sometimes you play to the line. And sometimes a little bit borderline," he said.
Tottenham were booed off at halftime and suffered another blow when striker Richarlison pulled up with a hamstring injury.
Frank said the injury was likely the result of the workload players faced over the Christmas and New Year period.
"I think we've done a lot of things right to get him where he is now. I think he's done a lot of things right. It's not only about us, it's about every club. I think there's a few injuries across the Premier League. I think it's something for the football authorities to look into as well," he said.
"We played five games in 13 days, four in ten. One of the few clubs who did that."