Republican Matt Bevin’s victory in the Kentucky governor’s race Tuesday highlighted the enduring power of public sentiment about the federal health law to energize GOP voters as the national parties prepare for the 2016 elections. The Affordable Care Act, also known as Obamacare, played a central role in the contest, which Mr. Bevin won resoundingly, 53% to 44%, against Democrat Jack Conway. The Republican pledged to dismantle the state’s health exchange—which earned praise for its relatively smooth launch—and to roll back or modify the expansion of Medicaid under the law.