Texas Sen. Ted Cruz, whose combative approach to politics has helped guide the Republican Party’s right wing, entered the 2016 presidential race on Monday, kicking off a primary-election debate about how aggressively conservative the GOP should be as it seeks to recapture the White House.

“I’m running for president,” Mr. Cruz said in a Twitter post, becoming the first major candidate of either party to enter the race and heightening his national visibility.

The announcement by Mr. Cruz marks the beginning of the primary election battle to define a Republican Party that is divided about the balance between ideology and pragmatism, and which is uncertain about who should lead it. His candidacy comes as recent polls indicate that none of the likely candidates has yet emerged as a bridge-building consensus choice among the party’s factions.

Mr. Cruz will be planting his flag on the far right flank of what is expected to become a crowded primary field spanning an ideological spectrum from centrists, such as New Jersey Gov. Chris Christie, to the libertarian Kentucky Sen. Rand Paul and social conservatives like Mike Huckabee, a former governor of Arkansas.

Mr. Cruz stands for a brand of ideological conservatism that contrasts with former Florida Gov. Jeb Bush, who has said a successful presidential candidate must be willing to “lose the primary”—that is, risk angering the party’s most conservative followers—to succeed with the more centrist electorate in the general election.