He wants Congress to end its separate ban on drilling, in order to reduce US dependence on oil imports.
"Now the ball is squarely in Congress' court," he said, adding that it was "time for action".
Mr Bush's move comes as high oil prices have pushed US petrol prices to more than $4 (£2) per gallon.
Mr Bush added: "The American people are watching the numbers climb higher and higher at the pump and they're waiting to see what the Congress will do."
His move will have no effect unless Congress acts as well.
On Friday oil prices - which have been rising steeply during 2008 - touched new highs, peaking above $147 a barrel.
US energy needs are set to be a key issue in November's presidential election.
Republican John McCain is in favour of offshore oil drilling, whereas his Democratic rival, Barack Obama, opposes it.
And congressional Democratic leaders oppose ending the drilling ban.
They point out that oil companies already have 68 million acres under government leases they can drill.
"This proposal is something you'd expect from an oil company CEO, not the president of the United States," said Senator Barbara Boxer, a Democrat and chairwoman of the Senate Environment Committee.
"The president is taking special-interest government to a new level and threatening our thriving coastal economy."
Since 1981, a congressional moratorium has prohibited oil and gas drilling along the east and west coasts and in the eastern Gulf of Mexico, an area accounting for some 80% of the US's outer continental shelf.
The executive drilling ban was issued in 1990 by the current president's father, President George H W Bush, and then extended by President Bill Clinton.
Since then offshore drilling and exploration have only been allowed in the western and central regions of the Gulf of Mexico plus parts of Alaska.
The federal bans were enacted in part to protect tourism and lessen the chance of oil spills washing on to beaches.
Mr Bush wants Congress to pass legislation to allow "responsible offshore exploration" and give states a say about whether drilling should be allowed.
But Mr Obama's campaign condemned the move, saying: "It would merely prolong the failed energy policies we have seen from Washington for 30 years."
It would be unlikely if Democrats in Congress backed a policy that contradicted their presidential candidate ahead of the November election.Republican contender John McCain, who changed his previous opposition to offshore drilling, said he thought the decision was a "very important signal" and that "states should continue to decide".