Republican strategists say the nomination is providing their candidates with another way to portray Democrats as out of touch by emphasizing Ms. Kagan’s support of a policy at Harvard Law School that prohibited the military from officially recruiting on campus because of the Pentagon ban on gay men and lesbians serving openly in the military.
Within hours of her nomination Monday by President Obama, Ms. Kagan’s stance was under attack by Republican candidates.
“As a member of a military family whose husband was in R.O.T.C. and served in the Iraq war, I am profoundly concerned about Kagan’s radical views regarding our military,” said Kelly Ayotte, a Republican Senate candidate in New Hampshire.
At the same time, Democrats saw opportunity as well.
In Illinois, Alexi Giannoulias, the Democratic Senate candidate, pressed his Republican rival, Mark Kirk, a House member who had voiced opposition to the nomination of Sonia Sotomayor, on whether he would also oppose Mr. Obama’s nomination of another woman to the court.
The exchanges show how even the nomination of someone who is considered likely to be confirmed can still reshape the electoral landscape. It is the first Supreme Court nomination in a midterm election cycle since 1994.
While some Republicans said they hoped the Supreme Court battle did not become so volatile that it overshadowed issues of the economy and jobs, party leaders said the nomination fight could be a galvanizing force for conservative voters.
“It’s a powerful reminder of how many seats we have in the Senate is important,” Ron Nehring, chairman of the California Republican Party, said in an interview Tuesday. “It also will raise a steady stream of issues that will be of great importance to many voters.”
The rise of Tea Party advocates, accompanied by protesters across the country waving copies of the Constitution as they demonstrate, underscores the degree to which voters could be aroused by debate at the confirmation hearings over interpreting the Constitution.
Democrats said they would mount an aggressive defense of Ms. Kagan that would neutralize any impact on the campaign season and maybe turn the confirmation proceedings to their advantage.
The White House, along with the Democratic National Committee, has created a 24-hour war room. Not an hour went by on Tuesday when the forces advocating on Ms. Kagan’s behalf did not fire off an e-mail statement or a video intended to either correct the record or call out their opponents.
“I have no doubt that Republicans want to gin it up for their base, but at the end of the day I think they are going to see a nominee who has the ability, the intellectual stamina and the right temperament to be on the court,” said Senator Robert Menendez of New Jersey, chairman of the Democratic Senatorial Campaign Committee.
Republicans say the import of a Supreme Court nomination could help them over all by consuming Senate time and attention, depriving Democrats of the chance to achieve other legislative victories.
“A Supreme Court nomination always stops things around here,” said Senator Lamar Alexander of Tennessee, the No. 3 Republican in the Senate. “It takes up most of the space for two or three months.”
He and several other Republicans pointed to the military recruitment issue as a potential obstacle to Ms. Kagan winning Republican votes. It not only allows Republicans to declare their support for the military but it also could conceivably be used against Democrats running in more conservative states — Senators Michael Bennet in Colorado and Blanche Lincoln in Arkansas, for instance — putting them on the spot over whether they agree with Ms. Kagan’s view on recruiting.
Rob Jesmer, executive director of the National Republican Senatorial Committee, said the idea of blocking recruiters from Harvard would not make sense to most Americans. “We are fighting two wars, and we can’t have people go recruit the best and the brightest?” he said. “It is elitist.”
Democrats say that Ms. Kagan’s views are being misconstrued and that she was mainly enforcing an existing ban while taking steps on her own to foster contact between Harvard and military recruiters.
“Her staunchest supporters are former military officers who are there at the Harvard Law School,” said Senator Jack Reed, Democrat of Rhode Island, an Army veteran and West Point graduate who invited Ms. Kagan to speak at the military academy. “She just wowed them.”
With other issues likely to surface, the confirmation battle adds one more element to an unsettled climate. While Republicans said they believed they stood to benefit from the fight, they also do not want it to overtake other topics.
Gentry Collins, political director of the Republican National Committee, said Tuesday that the hearings could temporarily become an issue in the midterm elections, but that most voters were more concerned about the economy and unemployment.
“People are worried about their jobs,” Mr. Collins said.