Initial signs had indicated the crash was an accident and that no foul play was involved.
President Kaczynski, his wife and dozens of Polish officials were among the nearly 100 passengers killed when their plane crashed while trying to land in western Russia.
Kaczynski and his delegation were on their way to commemorate the 70th anniversary of the massacre of thousands of Polish officers by Soviet secret police.
It was, said the Polish prime minister, the worst peace-time disaster in the country's history.
"This is one of the greatest tragedies in the history of our nation," Prime Minister Donald Tusk said. "The modern world has not seen a drama like this."
Eyewitnesses say the pilot attempted to land several times in heavy fog before crashing in a forest more than a mile from the airport in the Russian city of Smolensk.
While traffic controllers generally have the final word in whether it is safe for a plane to land, they can and do leave it to the pilots' discretion.
Air Force Gen. Alexander Alyoshin confirmed that the pilot disregarded instructions to fly to another airfield.
"But they continued landing, and it ended, unfortunately, with a tragedy," the Interfax news agency quoted Alyoshin as saying. He added that the pilot makes the final decision about whether to land.
The airplane, a Soviet-era Tupolev jet, has been involved in several recent crashes. Russian airline Aeroflot recently withdrew the model from service.
Reflecting the grave sensibilities of the crash to relations between the two countries, Russian Prime Minister Vladimir Putin personally assumed charge of the investigation. He was due in Smolensk later Saturday, where he would meet Polish Prime Minister Donald Tusk, who was flying in from Warsaw.
"This is unbelievable -- this tragic, cursed Katyn," Kaczynski's predecessor, Aleksander Kwasniewski, said on TVN24 television.
"On this difficult day the people of Russia stand with the Polish people," Medvedev said, according to the Kremlin press service.
In the Polish capital of Warsaw, mourners gathered outside of the presidential palace to show their grief, saying prayers, lighting candles and laying wreathes.
Kacynski, a one time advisor to former solidarity leader Lech Walesa, was considered a close U.S. ally.
In a written statement President Obama said: "Today's loss is devastating to Poland, to the United States, and to the world."
The crash devastated the upper echelons of Poland's political and military establishments. On board were the army chief of staff, the navy chief commander, and heads of the air and land forces. Also killed were the national bank president, deputy foreign minister, army chaplain, head of the National Security Office, deputy parliament speaker, Olympic Committee head, civil rights commissioner and at least two presidential aides and three lawmakers, the Polish foreign ministry said.
The deaths were not expected to directly affect the functioning of Polish government: Poland's president is commander in chief of its armed forces but the position's domestic duties are chiefly symbolic. Most top government ministers were not aboard the plane.
Poland's parliament speaker, the acting president, declared a week of national mourning.
Kaczynski's wife, Maria, was an economist. They had a daughter, Marta, and two granddaughters.
President Kaczynski, 60,is survived by several family members, including his twin brother, who once served as Poland's prime minister.
Poland announced Saturday that it would move up October presidential elections, despite the fact that two of the leading candidates -- President Kazcynski and an opposition leader who also was on board-- went down with the plane, along with the spirits of the nation.
(The Associated Press contributed to this report.)