Earlier, Republicans kicked off their national gathering by adopting a platform that endorses the president's agenda. Delegates also officially placed the names of Mr. Bush and Vice President Dick Cheney in nomination for a second term in the White House.
CBSNews.com webcast of the GOP convention.
"In choosing a president, we really don't choose a Republican or Democrat, a conservative or liberal," Giuliani said in prepared remarks that compared Mr. Bush with Ronald Reagan and Winston Churchill. "We choose a leader. And in times of danger, as we are now in, Americans should put leadership at the core of their decision."
Speakers on the convention's opening day mixed praise for Mr. Bush with attacks on his Democratic rival, Sen. John Kerry. House Speaker Dennis Hastert, R-Ill., called Kerry "weak on war and wrong on taxes. Folks, it's an easy choice."
While seeking to energize the party's conservative base with anti-Kerry rhetoric, Republicans also were reaching out to moderates who remain undecided. Most of the prominent convention speakers – Giuliani, McCain and California Gov. Arnold Schwarzenegger and Democratic Sen. Zell Miller – are popular among moderates and independents.
As the Monday evening session got under way, Vice President Cheney and his wife, Lynne, took their seats in the VIP box, to the cheers of delegates. And former President George H.W. Bush and his wife Barbara were welcomed with a video tribute.
Actor Ron Silver brought the delegates to their feet when he proclaimed, "The president is doing exactly the right thing."
Former New York Mayor Ed Koch, also a Democrat, delighted delegates by telling the morning session, "This year, I'm voting for the re-election of President George W. Bush." Koch was mayor from 1978-89.
Mr. Bush, meanwhile, campaigned in New Hampshire as the GOP opened its four-day gathering, and triggered an instant campaign stir when he told an interviewer he doubted victory is possible in the war on terror.
"I don't think you can win it. But I think you can create the conditions that those who use terror as a tool are less acceptable in parts of the world," he told NBC.
Democratic vice presidential candidate John Edwards quickly labeled that a concession of defeat in the war that terrorists launched in 2001.
"This is no time to declare defeat — it won't be easy and it won't be quick, but we have a comprehensive long-term plan to make America safer," he said of the Kerry campaign. "And that's a difference."
White House spokesman Scott McClellan hastened to clarify the president's remarks.
"He was talking about winning it in the conventional sense ... about how this is a different kind of war and we face an unconventional enemy," said the president's spokesman. "I don't think you can expect that there will ever be a formal surrender or a treaty signed like we have in wars past."
Earlier, McCain said it was fair game to criticize fellow Vietnam veteran Kerry's anti-war protests of three decades ago.
The Arizona senator told CBS' Early Show co-anchor Harry Smith that television ads accusing Kerry of lying about his military service in Vietnam were "dishonest and dishonorable." But, he added, "what John Kerry did after the war is very legitimate political discussion." Kerry was a leader of Vietnam veterans who opposed the war.
The convention began with polls showing the president and Kerry in a virtual tie. The first day was intended to focus on Mr. Bush's leadership in the war on terrorism, with a tribute to families of the Sept. 11 attacks.
Republicans roared voice-vote approval of a party platform that rallies behind Mr. Bush's agenda while endorsing positions dear to conservatives, including backing constitutional bans on gay marriage and abortion.
"Our platform highlights the principles that unite our party," Senate Majority Leader Bill Frist, the platform chairman, told delegates.
Republicans convened about four miles north of Ground Zero, where two hijacked planes destroyed both towers of the World Trade Center. Nearly 3,000 people died there, at the Pentagon and at a crash site in Pennsylvania.
Thousands of police kept the city under tight security as the convention opened. Normally bustling Penn Station was all but empty of travelers, with police officers posted every few feet.
Protesters, who numbered at least 120,000 during loud but peaceful demonstrations on Sunday, generally gave back the streets of Manhattan to commuters. One group of protesters in oversize Mr. Bush masks and wild costumes heckled convention-goers as they left their hotels Monday morning.
Inside the convention hall, delegates were exuberant about their candidate and about being in New York City, reports CBSNews.com's Joel Roberts.
"It's wonderful," said Paul Arndt, who traveled to New York from Lombard, Ill., with his wife, a Republican National Committee member. Arndt said the Republicans "appear to be gaining ground," and he predicted a "10-point bounce" for President Bush coming out of the convention.
Oscar Poole, a 74-year-old delegate from Georgia, said he's "feeling more secure all the time" after initially feeling "a little apprehensive" about coming to New York.
Dapper in a yellow suit and red-white-and-blue hat, Poole, who runs Col. Poole's Georgia Bar-B-Q in East Elllijay, Ga., said, "it saddens me that we have to have so much security here. But we need it. And that's why I'm for Bush."
He added that he was "pleased how friendly people are in New York City," and suggested that New Yorkers seem to have gotten friendlier since 9/11.