After his speech, Bush met with 30 pro-democracy activists, including Garry Kasparov, the former chess champion, who has called Russia a "police state".
There have been increased tensions between the Kremlin and Washington in recent weeks, and Putin said on Monday that Russia could take "retaliatory steps" if the US insisted on goiung ahead with the proposed missile defence system.
Polls show that more than 60 per cent of Czechs are against plans to station a radar system southwest of Prague, and surveys in Poland - where 10 interceptor missiles would be based - show strong opposition there as well.Cold War
Both countries are still negotiating with the US over whether to host the shield.
"My message will be: 'Vladimir - I call him Vladimir - you shouldn't fear a missile defence system"
Bush sought to play down Russia's response, which has overshadowed his European trip.
"The Cold War is over. It ended," Bush said at the medieval Prague castle, where he met Vaclav Klaus, the Czech president, and Mirek Topolanek, the prime minister.
"My message will be: 'Vladimir - I call him Vladimir - you shouldn't fear a missile defence system. As a matter of fact, why don't you co-operate with us on a missile defence system?" Bush said.
"The people of the Czech Republic don't have to choose between being a friend of Russia or a friend of the United States. You can be both."No threat
US experts contend the shield poses no threat to Russia because the missiles involved would be purely defensive and incapable of being fitted with warheads.
Bush said the system would be co-ordinated with Nato. He said he would urge Putin to participate.
"Please send your generals over to see how such a system would work. Send your scientists," Bush said.
Klaus said he had frank and open discussions with Bush on the missile shield, "and we understand each other".
"We are aware that the US bears high responsibility for the situation in the world, and I would like to stress that the United States and President Bush have our support in that," said Klaus, whose country has deployed troops to the US-led campaigns in Iraq and Afghanistan.
Bush also said he was working with the US congress to ease visa restrictions for Czechs.