"Russia is not the enemy," Bush said after meeting with Czech leaders in a visit en route to the G-8 summit in Germany. He said he would take a message to Russian President Vladimir Putin that "we can work together on common threats."
The Kremlin is bitterly opposed to the missile shield, and Putin warned on Monday that Russia could take "retaliatory steps" if Washington insists on building it.
Polls show more than 60 percent 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. Both countries are still negotiating with the U.S. over whether to host the shield.
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 with Czech President Vaclav Klaus and Prime Minister Mirek Topolanek.
"My message will be: 'Vladimir - I call him Vladimir - you shouldn't fear a missile defense system. As a matter of fact, why don't you cooperate with us on a missile defense 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," he 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 U.S. 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 U.S.-led campaigns in Iraq and Afghanistan.
Bush also said he was working with the U.S. Congress to ease visa restrictions for Czechs.
The European Union is pressing for visa-free travel for all 27 of its member states, including the Czech Republic, which joined the bloc in 2004.
The face of USA's First Lady Melania Trump after her handshake with Russian President Putin has received a lot of attention in social media
Moscow is trying to stop Balkan countries from entering NATO. Greece eventually took measures against Russia, even though Greek Prime Minister Alexis Tsipras had earlier said that Russia was Greece's strategic partner
The Ukrainian government refuses to abode by its obligations, rejects a peaceful resolution of the conflict, and disregards its own people, the president said