Romania’s president fought to keep his job Sunday, as voters cast ballots in a national referendum aimed at ousting him amid a bitter partisan fight that critics say has raised doubts about the rule of law in this young Eastern European democracy.

Both sides claimed victory after polls closed, and the outcome hinged on whether turnout exceeded the 50% threshold for the plebiscite to be legally valid. An initial estimate by election officials indicated it fell short. Final results were expected Monday.

President Traian Basescu, a center-right politician who had called on his supporters to boycott the vote, said Sunday …