CLEVELAND — Rachel Hoff on Monday became the first openly gay person to sit on the Republican Party’s Platform Committee.

But, she would not make history a second time. Facing overwhelming opposition and fighting back tears as she spoke to the committee, Ms. Hoff, a delegate from the District of Columbia, offered an amendment of a few paragraphs to the Republican platform that would have encouraged a “thoughtful conversation” within the party on same-sex marriage.

It received only about 30 votes from the 112-member committee, according to an unofficial count.

Ms. Hoff’s amendment did not call for the party to embrace same-sex unions. It called for an acknowledgment that a growing number of Republicans are changing their views on the issue and that opinions on marriage are “diverse and sincerely held.”

As Ms. Hoff addressed the committee, her voice broke. “We are your daughters, your sons, your friends, your neighbors, your colleagues,” she said. “All I ask today is that you include me and those like me.”