Republicans desperate to stop Donald J. Trump from capturing the presidential nomination increased the pressure Wednesday on Gov. John Kasich of Ohio to quit the race, with Jeb Bush joining the growing number of party figures throwing their weight behind Senator Ted Cruz.

Mr. Kasich refused, saying that he, not the Texas senator, was the best option to stop Mr. Trump. But his argument was undercut by his dismal showings Tuesday in Utah and Arizona, where he won no delegates — as well as by the surprise endorsement Wednesday morning by Mr. Bush of Mr. Cruz.

Mr. Bush, who dropped out of the presidential race last month, is the latest mainstream Republican — following Mitt Romney and Senator Lindsey Graham — who is ideologically closer to Mr. Kasich, but whose embrace of Mr. Cruz is a strategic calculation that he has a better shot at stopping Mr. Trump.

In a statement, Mr. Bush called Mr. Cruz a “consistent, principled conservative who has demonstrated the ability to appeal to voters and win primary contests.”