Bush also expressed his country's support to Croatia on its road towards the European Union and NATO.
Congratulating Mesic on being re-elected Croatian president, Bush hailed growing friendship between the two countries.
I am looking forward to Croatia's taking final steps in the continuation of its EuroAtlantic integration. In this spirit the United States is continuing to support progress which Croatia is making on the road towards membership of the EU and the North Atlantic Alliance, Bush said in his message.
I am looking forward to cooperating with You on this and many other important issues. I am confident that in the coming months and years partnership between our two countries will strengthen and deepen, the US President said.
According to preliminary results of Sunday's election run-off, the incumbent President Stjepan Mesic won 66 percent of votes and his rival, Deputy Prime Minister Jadranka Kosor, 34 percent.