The US, Australia, Japan and India have formed an international coalition to lead aid efforts after the Indian Ocean sea surges, the US president has said.
Aid groups are facing a challenge reaching the most remote areas
Mr Bush said the giant waves that swept ashore in 11 countries had brought loss and grief almost beyond comprehension.
The US has already pledged $35m and sent its navy to help the aid effort.
But a US senator has criticised the amount of aid pledged as paltry, saying it was the country's responsibility to shoulder the bulk of aid needs.
"We spend more than $35m every morning before breakfast in Iraq," Patrick Leahy, a Vermont Democrat told the BBC.
"I felt that $35m is almost a reflection that we are not paying attention to the magnitude of this terrible tragedy".
He proposed that the United States and other wealthy nations write a "blank cheque" to guarantee to pay for the immediate humanitarian and longer-term reconstruction needed in for the countries devastated by the quake-triggered waves.
"The Bush administration has written a very large blank cheque in Iraq," he said adding that the vast majority of Americans would back such a pledge.
The United Nations estimates that five million people need help to survive, while the total number of dead is put at more than 114,000.
UN relief co-ordinator Jan Egeland said it would take at least another two or three days to begin to respond properly to "the tens of thousands of people who would like to have assistance today - or yesterday."
Outgoing US Secretary of State, Colin Powell, told the BBC the US-led group of nations would co-ordinate its work with the UN and other relief groups to ensure aid went to areas where it was most desperately needed.
"I would expect that in a short period of time this core will expand and the entire international community will be involved," he said.
He said the four countries had come together because of their proximity to the disaster zone.
Mr Bush backed calls for a tsunami alert system after what he called a "terrible disaster" and predicted other countries would join the "core group".
The US state department said on Tuesday that 12 Americans were among the 77,000 people so far known to have died. Hundreds of Americans remain missing.
The National Geospatial-Intelligence Agency is using spy satellites to assess damage to roads, bridges, ports and airfields and passing the information to US agencies handling disaster relief.
'World will cope'
The sea surges that swept parts of Indonesia, India, Sri Lanka, Thailand and other coastal areas around the Indian Ocean represented "one of the major natural disasters in world history", Mr Bush said.
The president said he had talked to the leaders in the affected region and was working to target initial relief efforts to the things most required.
"Together the world will
cope with their loss. We will prevail over this destruction," Mr Bush
The US pledged military personnel and long-term rebuilding assistance.
Mr Bush said he would consider all requests for aid to countries affected, including a
German proposal for a moratorium on debt repayment for Indonesia and Somalia.
The president also dismissed as "misguided" a United Nations official's initial suggestions that rich nations were not giving enough for disaster relief.
UN emergency relief co-ordinator Jan Egeland said on Tuesday that his comments had been misunderstood.
"I have been misinterpreted when I yesterday said that my belief that rich countries in general can be more generous," Mr Egeland said.
"This has nothing to do with any particular country or the response to this emergency. We're in early days and the response has so far been overwhelmingly positive."
But the New York Times said Mr Egeland had been right to suggest that US foreign aid contributions were too small.
"$35 million remains a miserly drop in the bucket, and is in keeping with the pitiful amount of the United States budget that we allocate for non-military foreign aid," the newspaper said in an editorial on Thursday.