CLEVELAND — Same-sex marriage and transgender rights are emerging as points of serious strain between social conservatives and moderates who are trying to shape the Republican platform, reviving a festering cultural dispute as thousands of party activists and delegates prepare for their convention.

Caught in the middle is Donald J. Trump, who claims “tremendous support, tremendous friendship” from gays, lesbians, bisexuals and transgender people, and has gone further than most party figures to embrace them. Gays, in fact, are one of the few minority groups Mr. Trump has not singled out for criticism. But as the presumptive Republican nominee, he is also trying to assuage doubts about the convictions of his conservatism.

The uncomfortable dynamic Mr. Trump has created for himself is perhaps best illustrated by his own calendar.

He huddled last month at a Manhattan hotel with hundreds of religious conservatives, many of them — like James C. Dobson, the founder of Focus on the Family, and Tony Perkins, the president of the Family Research Council — outspoken opponents of new legal protections for gay and transgender people.