"Now we must extend a hand of opportunity to those who seek peace. As part of a lasting ceasefire, Gaza's border crossings should be open to allow the flow of aid and commerce," Obama said.
Obama also reiterated the US backing for international demands made of the Hamas faction that governs Gaza: that it recognise Israel, end violence and agree to recognise previous peace agreements with Israel.
He also said the US would support efforts to end weapons smuggling across the Gaza border from Egypt.
Obama did, however, call for a complete withdrawal of Israeli forces from Gaza following its offensive, and said the US would provide humanitarian and economic assistance to the Palestinians living in the Gaza Strip.
"It seems that Obama is trying to repeat the same mistakes that [former US president] George Bush made"
Osama Hamdan, a Hamas spokesman
Osama Hamdan, a Hamas spokesman, told Al Jazeera that Obama's remarks seemed to show that the US viewed the situation through "Israeli eyes".
"It seems that Obama is trying to repeat the same mistakes that George Bush made without taking into consideration Bush's experience that resulted in the explosion of the region instead of reaching stability and peace in it," he said.
"I think this is an unfortunate start for President Obama in the region and the Middle East issue. And it looks like the next four years, if it continues with the same tone, will be a total failure."
More than 1,300 Palestinians, and 13 Israelis, died after Israel began its offensive in Gaza on December 27, before both sides separately declared a ceasefire on January 18.
Mitchell, 75, acknowledged there were "many reasons to be sceptical" about the prospects for peace between Israelis and Palestinians.
"The key is the mutual commitment of the parties, and the active participation of the United States government," he said.
He is best known for helping to broker Northern Ireland's landmark Good Friday agreement in 1998 which ended decades of bloody conflict, and experience which he said led him to form the conviction that "there is no conflict that cannot be ended".
In 2000, he also presided over a committee investigating the ongoing violence of the Middle East conflict, which recommended Palestinians do more to stop attacks on Israel and Israel stop settlement building on occupied land.
Obama also named Richard Holbrooke, a former UN ambassador, as his special envoy to Afghanistan and Pakistan.
Holbrooke, another US political veteran with years of diplomatic experience, is best known for brokering the 1995 peace agreement that ended three years of war in Bosnia.
Holbrooke, on accepting the position, said he had been given a "daunting assignment".
Journalist and analyst Roy Gutman told Al Jazeera that the fact that Obama appointed Holbrooke as the first such envoy for the region on only his second day in office shows that the issue is a top priority for the Obama administration.