Washington (CNN) -- One day after she claimed victory in the nation's last unresolved Senate race, Sen. Lisa Murkowski of Alaska reiterated Thursday that she does not support Sarah Palin as the Republican presidential nominee in 2012.
Murkowski was defeated in this year's primary by fellow Republican Joe Miller but apparently won a long shot write-in bid for the Senate seat she currently holds, finally claiming victory more than two weeks after the Nov. 2 elections. Palin, a former Alaska governor and 2008 vice presidential candidate, had backed Miller.
Palin told Barbara Walters on Wednesday she could beat President Obama if she was the GOP nominee in 2012.
"She is clearly setting her sights on that higher office," Murkowski said on CNN's "American Morning" program.
"I have in the past said that Sarah Palin would not be my candidate," Murkowski added, declining to name anyone she would support.
"I'm still looking to see who will emerge in the Republican field of candidates. And I think that remains to be seen."
Murkowski clearly distanced herself from Palin, who had defeated her father, Frank Murkowski, in the 2006 GOP gubernatorial primary. The senator was first appointed to her post by her then-governor father in 2002.
"There's been a lot of assumptions that, because, well, because two women are from Alaska we should be friends and supporters automatically," Murkowski said. "We are two very different women and are choosing two very different paths."
Murkowski also reflected on her relationship with the Republican Party, which opposed her candidacy after her defeat in the August primary.
"I was not my party's nominee," she said. "I went outside the box."
But Murkowski made it clear she remains a Republican and will continue to conference with GOP senators.
Still, Murkowski said, she was elected by a coalition of Alaskans, and "it will be all Alaskans that I will be representing."
Murkowski declared victory Wednesday, saying the result of her write-in candidacy was a "miracle."
"Against all odds, we as Alaskans together made history," Murkowski told cheering supporters in Anchorage.
If she prevails in a potential challenge, Murkowski would become the second person to win a write-in bid for the U.S. Senate.
Strom Thurmond of South Carolina won a write-in campaign in 1954.
Murkowski's victory claim came after a bitter and prolonged Senate battle in which she fended off a three-pronged attack: from Miller, the Tea Party Express political movement and Palin.
"We've said all along, we'll wait for the votes to be counted," Murkowski campaign manager Kevin Sweeney told CNN.
"By any standard, this was a clear victory. This was a whuppin'," Sweeney added.
Alaska's Division of Elections has tallied 100,868 votes for Murkowski and 90,448 for Miller. By that count, Murkowski leads by 10,420.
However, the MIller campaign has challenged 8,153 of Murkowski's votes. If, as Miller's side would like, all of those contested votes were thrown out -- an almost impossible scenario to many election and legal observers -- Murkowski would still win the race with the uncontested votes alone.
"The numbers are what the numbers are," Murkoswki said Thursday. "We are clearly, clearly ahead. The numbers just don't add up for Mr. Miller."
Miller has not conceded.
In an interview with Neil Cavuto on Fox News, Miller said he is "less cautiously optimistic" given the vote tallies. But Miller was emphatic about ensuring what he called the integrity of the vote-counting process, asserting that the difference was "less than 1 percent."
He said he may request a recount, that he wants to ensure that some military ballots were mailed out and that the difference between his votes and Murkowski's may shift if a "consistent standard" for evaluating the write-in ballots is applied.
Evaluating the write-in ballots has been controversial. As Alaska's Division of Elections tallied them, some ballots that contained misspellings or variations of "Lisa Murkowski" have been credited as votes for her.
That issue of evaluating a "voter's intent" has infuriated the Miller campaign, prompting it to launch a federal lawsuit in hopes of annulling the count of misspelled or incongruous ballots. Miller's campaign blasted the Division of Elections' standards as "extraordinarily ambiguous."
A Miller loss would be a huge slap to Palin and the Tea Party Express. Palin endorsed Miller over Murkowski for the state's GOP Senate primary, and the national Tea Party group followed suit. Murkowski lost the primary in August -- and conceded the race. Shortly after, Murkowski launched a write-in bid to retain her seat.
CNN's Shannon Travis and Kristi Kreck contributed to this report.