The battle for presidential votes in the key state of Ohio is heading to court.

President Obama's campaign has sued the state of Ohio over new rules for early voting designed to benefit members of the military, saying the extra hours should be available to all voters.

That lawsuit prompted claims by Mitt Romney and aides that the Obama campaign is targeting military voters -- a false claim, Obama's team quickly responded.

Said Romney in a statement: "President Obama's lawsuit claiming it is unconstitutional for Ohio to allow servicemen and women extended early voting privileges during the state's early voting period is an outrage."

Rob Diamond, director of veterans and military family voting for the Obama campaign, said Romney is fabricating his claim: "The Obama campaign filed a lawsuit to make sure every Ohioan, including military members and their families, has early voting rights over the last weekend prior to the election."

Obama political adviser David Axelrod, appearing on Fox News Sunday, said Romney's claims are "false and misleading," and that the lawsuit is about "whether the rest of Ohio should have the same right" to early voting as members of the military.

"I think it's shameful that Governor Romney would hide behind our servicemen and women to try and win a lawsuit to deprive other Ohioans of the right to vote," Axelrod said.

Ohio -- with 18 electoral votes -- is not just any member of the Electoral College; it has picked the winner in 12 straight presidential elections.

The Buckeye State figures to be particularly important for Romney; no Republican has won the presidency without taking Ohio.

The lawsuit reflects the intense ground battle between Obama and Romney in up to a dozen battleground states. Both campaigns, plus outside groups, are grappling state by state, county by county, precinct by precinct.

ABC News notes that the flap over Ohio stems from the nationwide battle over voter qualifications: