Michigan’s Republican presidential primary Tuesday is more than a test of Mitt Romney’s ability to win his native state. It’s a test of a central thesis of his campaign: He has the best chance of beating President Barack Obama.

The results will show whether Mr. Romney—the longtime GOP front-runner now locked in a close race nationally with former Sen. Rick Santorum—can win the blue-collar voters who are a pivotal bloc in the Midwest, where parties win or lose presidential elections.

And more than any other state so far, Michigan will demonstrate whether Mr. Romney can appeal to the party’s core social conservatives and tea-party supporters who have resisted him in prior contests.

The fallout will reach beyond Michigan to the campaign’s single biggest day next Tuesday, when Republicans in 10 states vote in Super Tuesday primaries and caucuses.