During their weekend summit, the two leaders were unable to resolve differences over Clinton's proposal for a missile defense system. Implementation of such a system would require changes in the 1972 Anti-Ballistic Missile (ABM) Treaty.
Both Republican Bush and Al Gore, his Democratic presidential opponent, separately reiterated that they would scrap the treaty if Moscow doesn't eventually go along.
That is essentially the current administration view, although Clinton and Gore see such a move as a last resort.
Bush welcomed the Clinton-Putin agreements reached over the weekend in which Russia and the United States will reduce stockpiles of weapons-grade plutonium and establish an early warning center in Moscow to exchange information on the launching of ballistic missiles.
"I thought that was OK, that was fine," Bush told reporters after delivering a D-Day commemorative speech on the banks of the Savannah River. "That does not necessarily tie a future president's hands."
But the Texas governor said the United States should forge ahead with developing a missile defense that would be more robust than the limited system the administration is considering - with or without Moscow's approval.
"We've got to be able to have a missile defense system that adequately addresses the new threats facing America," Bush said. "That's why we need to have the treaty fully amended or withdraw from the treaty."
Earlier, Vice President Gore said he might be willing to pull out of the ABM Treaty if Russia doesn't eventually agree to amendments that would allow a limited missile defense umbrella.
"I wouldn't rule that out," Gore said.
The vice president said there was a big difference between his stance and "tearing up the treaty and throwing it away." He said he wants a treaty that would allow "a modest, affordable and limited system that's designed to protect us against one of these rogue states that might have a handful of weapons, five, 10, 15 years from now."
Gore had said on Monday that he believes Russia will eventually agree to modify the treaty to allow a limited U.S. missile defense system.
In Georgia, Bush - on the 56th anniversary of D-Day - proposed a more responsive Veterans Affairs system, a national memorial in Washington, D.C., and a rebuilt military aimed at improving morale.
The governor, whose father - former President Bush - was a Navy pilot in World War II, also paid tribute to the 6,600 Americans killed in the Normandy landings and to those who survived.
"Within one year of June 6, 1944, Europe and its concentration camps were liberated, and the world was saved from Nazi tyranny, and that's why we're here today," Bush tod about 300 people at Hero's Overlook.
He said the Veterans Affairs Department is fighting a 500,000-case backlog and he, as president, would work to clear that.
"The whole process has been rife with errors and oversights and even missing records," he said.
Bush also endorsed the idea of building a national World War II memorial in Washington.
The GOP candidate accused Mr. Clinton and Gore of undercutting morale in today's military with weapons cutbacks and too many overseas deployments.
"After seven years, the Clinton-Gore administration has allowed the morale to slip to dangerously low levels. I will rebuild the military power of the United States in order to keep the peace," Bush said.