Democrats moved to make same-sex marriage a part of their party platform, placing language that would declare a right for gay men and lesbians to marry on track for approval by the party’s leadership. Party officials met over the weekend in Minneapolis and approved the first step in the platform-amending process. In two weeks, the entire platform committee will vote on the matter at a meeting scheduled in Detroit. Then, if approved as expected, it would move on to convention delegates in Charlotte, N.C., for final approval in September. According to Democrats who were briefed on the vote in Minneapolis,...