United have won three of their last 12 matches in all competitions and are seventh in the Premier League standings, six points off fourth place and the final Champions League spot with seven games left.
Rangnick joined United on an interim basis in November following the sacking of Ole Gunnar Solskjaer and has led the team to nine wins in his 22 matches in charge so far.
"No, no regrets whatsoever," he told reporters ahead of Saturday's home league game against Norwich City. "I would do it over and over again. I don't regret that at all.
"As a manager, you always have to question what you could have done better but I have no regrets ... We are all aware it's not an easy squad and the whole situation was not easy or I would not be sitting here and Ole would probably still be here.
"We have shown in the past couple of months we were able to raise the level but not in the same sustainable way that I would have wished. That's the reason I'm not happy at all with what we've achieved so far."
Rangnick, 63, will be interim manager until the end of the season before taking up a consultancy role with the club for a further two years.
United are searching for a permanent manager, with British media reporting that Ajax Amsterdam boss Erik ten Hag is the likely candidate for the top job.