The League last month sued Scott's chief elections official, Secretary of State Ken Detzner, whose office in 2014 interpreted state law to exclude state university buildings from a list of sites available for early voting.
Florida allows early voting at elections offices, city halls, libraries, fairgrounds, civic centers, courthouses, county commission buildings, stadiums, convention centers, government-owned senior centers and government-owned community centers.
But buildings on state college and university campuses? No. Democrats tried to include them as early voting sites, but Republicans blocked the proposal.
Scott's lawyers asked the federal court to step aside and let the case be decided by a state judge.
"A state court, interpreting state law, can decide the case on narrow, statutory interpretation grounds and, perhaps, avoid any constitutional issues," the state's brief said.
The case is assigned to U.S. District Judge Mark Walker in Tallahassee, an appointee of former President Barack Obama, who has ruled decisively against Scott in two previous voting rights cases.
The plaintiffs in the lawsuit are nine students at UF and Florida State.
One of them, Megan Newsome, an astrophysics major, wrote an op-ed column in the Gainesville Sun last November that got the attention of Guy Cecil, the head of a Democratic super PAC, Priorities USA, which supports the lawsuit against Scott's administration.
Scott's legal brief dismissed claims by students that they faced parking and logistical hurdles that makes it harder for them to vote early.
"Having to go one mile off campus for college-aged voters who have a history of actually voting in elections cannot impose a constitutionally cognizable burden or serve as the basis for a plausible claim for relief," Scott's lawyers wrote.
The lawsuit stems from a decision by Detzner's office in 2014 to reject a request by the city of Gainesville to use the Wayne Reitz Student Union building at UF as an early voting site.
With the student union off-limits for early voting, the closest early voting site for UF students is the county elections headquarters, which is 1.2 miles from campus.
As governor, Scott signed into law restrictions on voting that Democrats claimed was a voter suppression tactic to depress liberal turnout for the 2012 election when Obama sought re-election.
Scott signed a subsequent law in 2013 that expanded the hours, days and locations of early voting. The Reitz Union can be used as a polling site on election day. The question in the suit is whether it can be used for early voting, which is growing in popularity throughout the state.