CLEVELAND — Rancor and hard-edged attacks dominated the start of the Republican National Convention on Monday as speakers branded Hillary Clinton as a liar who deserved to be in prison and two African-American Republicans ridiculed the Black Lives Matter movement.

The divisive day even swept up the presumptive nominee, Donald J. Trump. Renegade delegates forced a floor fight in an effort to embarrass him, and his top aide called Ohio’s governor “petulant” for not endorsing Mr. Trump.

The House speaker, Paul D. Ryan, who is the chairman of the convention, also distanced himself from Mr. Trump’s approach to racial unrest and from one of his signature proposals.

The unusual jousting among Republicans at their own convention gave way to more traditional, fiery speeches aimed at Democratic leaders, Mrs. Clinton and President Obama. The most impassioned remarks came from former Mayor Rudolph W. Giuliani of New York, who described an America plagued by crime and violence and repeatedly praised police officers for keeping families safe without regard to race.