"I look forward to working with the Senate in the next stage of this process, and I thank you again, Mr. President, for this honor of a lifetime," she said.
Republicans are expected to criticize her for attempting to bar military recruiters from the Harvard Law campus while she was dean. That issue was used against her by critics during her confirmation hearing last year for her current post.
With control of 59 votes in the Senate, Democrats should be able to win confirmation. However, if all 41 Republicans vote together, they could delay a vote with a filibuster.
Republicans have shown no signs in advance that they would try to prevent a vote on Kagan, but they are certain to grill her in confirmation hearings over her experience, her thin record of legal writings and her decisions at Harvard.
The senator who will preside over her confirmation hearing, Democrat Patrick Leahy of Vermont, said, "The Senate should confirm Ms. Kagan before" Labor Day.
"Our constituents deserve a civil and thoughtful debate on this nomination, followed by an up-or-down vote," said the chairman of the Senate Judiciary Committee.
The Republican Senate leader, Mitch McConnell of Kentucky, said his party would make sure there was a "thorough process, not a rush to judgment" on the nomination.
"Judges must not be a rubber stamp for any administration. Judges must not walk into court with a preconceived idea of who should win," he said, adding that Republicans would have a vigorous debate on that principle.
Obama introduced Kagan as "my friend."
"Elena is widely regarded as one of the nation's foremost legal minds. She's an acclaimed legal scholar with a rich understanding of constitutional law. She is a former White House aide, with a life- long commitment to public service and a firm grasp of the nexus and boundaries between our three branches of government," Obama said.
Kagan served in the Clinton White House.
Obama began with high praise for the retiring Stevens, a leader of the court's liberals, calling him "a giant in the law," impartial and having respect for legal precedence.
Kagan "embodies the same excellence, independence and passion for the law," Obama said.
He noted that neither Kagan's mother nor father "lived to see this day, but I think her mother would relish this moment. I think she would relish, as I do, the prospect of three women taking their seat on the nation's highest court for the first time in history ... a court that would be more inclusive, more representative, more reflective of us as a people than ever before."
Kagan praised Stevens for having "played a particularly distinguished and exemplary role. It is therefore a special honor to be nominated to fill his seat."
In a short tenure as solicitor general "I have felt blessed to represent the United States before the Supreme Court, to walk into the highest court in this country when it is deciding its most important cases, cases that have an impact on so many people's lives," she said.
Seven Republicans voted for her confirmation last year as solicitor general.
One of them, Orrin Hatch of Utah, a member of the Judiciary Committee, issued a statement saying his decision this time "will be based on evidence, not blind faith. Her previous confirmation and my support for her in that position do not by themselves establish either her qualifications for the Supreme Court or my obligation to support her."
Kagan would become the only justice who had no prior experience as a judge. The other justices all served previously as federal appeals court judges. She was nominated to a federal appeals court post by President Bill Clinton but never served.
That means Kagan has a smaller paper trail than other recent nominees since there are no prior decisions to scrutinize.
But conservatives were already mounting an attack, one they laid the groundwork for when she was mentioned last year as being on Obama's short list for the Supreme Court post last time around.