MERRIMACK, N.H. — Former South Bend Mayor Pete Buttigieg on Tuesday said his performance in the New Hampshire primary proves his campaign is “here to stay” as he called for party unity while lobbing unnamed attacks directed toward his Democratic presidential rivals.

“Thanks to you, a campaign that some said shouldn’t be here at all has shown that we are here to stay,” Mr. Buttigieg told supporters in Nashua.

He congratulated Sen. Bernard Sanders of Vermont on a “strong showing” in New Hampshire. As he spoke, multiple TV networks projected Mr. Sanders as the winner of the New Hampshire primary, with Mr. Buttigieg a close second place.

Mr. Buttigieg offered an olive branch by noting that his supporters chanted “Vote blue, no matter who” earlier in the day at Sen. Elizabeth Warren of Massachusetts.

“We are on the same team,” he said.

But he went on to knock a “my way or the highway” approach — a label he and his campaign have used before in referring to Ms. Warren.

“A politics of my way or the highway is a road to re-electing Donald Trump,” he said, saying vulnerable Americans don’t have the luxury of pursuing “ideological purity.”

“In this election season, we have been told by some that you must either be for a revolution or you are for the status quo. But where does that leave the rest of us?” he said, repeating a version of a line that prompted jeers from Sanders supporters at a major party dinner over the weekend.

He also said some might dismiss him as a “naive newcomer.”

“But a fresh outlook is what makes new beginnings possible,” he said.

Sen. Amy Klobuchar of Minnesota, who was on track to finish third in New Hampshire, used the phrase “cool newcomer” when talking about Mr. Buttigieg at Friday’s debate.

• Seth McLaughlin reported from Washington.

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