The Glasgow giants have lost four consecutive games for the first time since 1978, in the Frenchman's first four matches.
Just two weeks into the role, Nancy faced calls from the visiting fans at Tannadice to be sacked.
Earlier in the day, Celtic chairman Peter Lawwell announced he will depart at the end of the year due to "intolerable" levels of "abuse and threats" from disgruntled fans.
After winning 13 of the last 14 Scottish Premiership titles, Celtic have endured a torrid season on and off the pitch, with fans calling for changes in the boardroom.
Another damaging defeat leaves them six points adrift of league leaders Hearts and just three clear of rivals Rangers.
After losing to Hearts, Roma and in the League Cup final to St Mirren on Sunday, Nancy looked set to finally turn the corner during a dominant first half.
Daizen Maeda opened the scoring for Celtic, who should have been further clear at the break but for a series of missed chances by Johnny Kenny.
However, the match swung decisively in United's favour in three minutes either side of the hour mark.
Krisztian Keresztes equalised before Zac Sapsford curled brilliantly past Kasper Schmeichel.