CAMBRIDGE, Mass. — Only one Republican holds a seat in the House from all of New England. Other than that single district in Maine, an area of farms and fishing villages and forests that sent an electoral vote to Donald Trump in 2016, the rest of the region is a vast swath of blue.

But the fate of New England’s lone House Republican, Representative Bruce Poliquin of Maine, grew uncertain this week as the margin between Mr. Poliquin and his closest opponent remained narrow and Maine officials went on counting votes a week after they were cast. This wasn’t a recount, but a new and complex method of voting that Maine residents chose, and one that now had become the center of Mr. Poliquin’s dilemma — and ire.

By Tuesday, Mr. Poliquin, a two-term incumbent, was suing Maine’s secretary of state, calling for a halt to the counting and, in essence, asking the state to take its new voting system and toss it. Under a regular voting system, Mr. Poliquin asserted, he would have won.

Neither Mr. Poliquin nor his Democratic opponent, Jared Golden, won a majority of the votes on Election Day. No final and official tally was released, although numbers cited by the Maine Secretary of State’s office last week showed Mr. Poliquin with a narrow lead after many of the votes were counted. Under Maine’s new system, known as ranked-choice voting, since no one had an outright majority of votes, a new tally was triggered in which the second and, possibly, third choices of some voters will be added to the count.