WASHINGTON — Warning that beleaguered Democrats cannot afford a replay of their contentious presidential primary in the race to lead the Democratic National Committee, the mayor of South Bend, Ind., Pete Buttigieg, entered the contest for party chairman on Thursday, presenting himself as an alternative to the two leading candidates.

“This is not a time to relitigate an old battle,” Mr. Buttigieg said. “We’ve got to transcend the narrative that this is some kind of proxy fight.”

He did not mention them by name, but the 34-year-old mayor was referring to Labor Secretary Thomas E. Perez and Representative Keith Ellison of Minnesota, who have secured the most support to date. Mr. Perez was an early supporter of Hillary Clinton and a finalist to be her running mate while Mr. Ellison was one of Senator Bernie Sanders’s most visible surrogates.

Many Democrats have expressed concern about reopening wounds from a presidential primary that highlighted their ideological, generational and racial fissures. Mr. Buttigieg supported Mrs. Clinton late in that race, but he is now casting himself as someone who can rise above the fierce internecine wars that linger from 2016, presenting himself as an alternative to a pair of Washington-based candidates.