Citing Carter's favorite song — "Reach Out, I'll Be There," by the Four Tops — Obama said: "Ash, I'm reaching out to you ... You have been there."
Carter pledged to work hard for the nation's armed forces and told the president, "If confirmed in this job, I pledge to you my most candid, strategic advice."
Requesting quick Senate confirmation of Carter, Obama pointed out that his nominee has been confirmed for three previous jobs and "he's respected and trusted on both sides of the aisle."
Republicans, who will begin control of the Senate in January, generally praised the nominee, as did Democrats.
Sen. Jack Reed, D-R.I., a member of the Senate Armed Services Committee, which will hold hearings on Carter's nomination, said "his command of national security policy and his commitment to our men and women in uniform will serve him well as our next Secretary of Defense."
If confirmed, Carter will replace Defense Secretary Chuck Hagel, who announced his resignation last month.
Hagel did not attend the ceremony but issued a written statement endorsing Carter's nomination and calling him "a patriot and a leader."
Nebraska Sen. Deb Fischer, a Republican member of the Armed Services Committee, praised Carter's "extensive experience" but said she is concerned that the Obama administration "continues to lack a coherent strategy to counter an array of global crises."
They include Russian aggression in Ukraine and the fight against the Islamic State in Iraq and Syria.
Carter served as deputy secretary of Defense from October 2011 until his retirement in December 2013, with specialties that include science, technology and developing the Defense budget in tight economic times.
From 2009 to 2011, Carter — a Yale graduate with a degree in physics — worked as undersecretary for acquisition, where he supervised the procurement of equipment to meet emerging threats.
Before he left the Pentagon, Carter was the military's top weapons buyer and pushed hard for gear that troops needed to stay alive on the battlefields in Iraq and Afghanistan: Mine Resistant Ambush Protected (MRAP) vehicles to protect them from roadside bombs, surveillance equipment to spy on insurgents and bomb-sniffing dogs to find the buried mines that killed and maimed troops on foot patrols.
Regarded as smart, capable and wonkish, Carter also led the Pentagon review of its budget with a look toward cuts.
In fiscal year 2001, adopted before the 9/11 terrorist attacks, the Pentagon's budget was $297 billion. Ten years later, it had ballooned to $687 billion, including funding for the wars in Iraq and Afghanistan.
At the Pentagon on Thursday, Hagel described his resignation as "a mutual decision" based on private talks he had with Obama. The outgoing secretary said the Pentagon faces new challenges and that he considers the president a friend.
"Friends can talk plainly to each other," Hagel said. "I think you have to know when to leave, too.'"
At the White House, spokesman Josh Earnest said the next Defense secretary's challenges will range from the battle against the Islamic State to the fight against Ebola to efforts to reduce sexual assaults among military personnel.
Asked about reports that some Pentagon officials feel micro-managed by the White House, Earnest said the military has bigger problems with Congress because it refuses to pass needed budget reforms.
Tensions between the White House and Pentagon are common to all presidencies, Earnest said, but the bottom line is civilian control of the military.
The next Defense secretary "will be very clear about what the chain of command is," Earnest said, and understand that "the president of the United States is the commander in chief and sits at the top of the chain of command. That means the president bears significant responsibility for what happens at the Pentagon and the Department of Defense."
