As part of an international charm offensive, British Foreign Secretary Philip Hammond was to meet Israeli Prime Minister Benjamin Netanyahu, who strongly opposes the landmark accord stuck on Tuesday in Vienna between Iran and world powers led by Washington.
Netanyahu has condemned it as a "historic mistake" and hinted at a possible military response.
In Washington, Secretary of State John Kerry was to hold talks with Saudi Foreign Minister Adel al-Jubeir, whose government has also been alarmed about the deal with its regional rival.
As the freshly-inked deal was put to members of the UN Security Council, a combative Obama said opponents at home and abroad had offered only a path to war.
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The issue is either resolved "diplomatically, through a negotiation, or it's resolved through force. Through war. Those are the options."
Obama's Republican rivals, who hope to scupper the agreement in a planned Congressional vote, have accused him of appeasement.
The president has said he will veto any attempt to block the deal.
Obama also addressed the concerns of Saudi Arabia and other Sunni Gulf states that the accord legitimises what they see as Iranian interference in the oil-rich region.
The agreement would not end "profound differences" with the Shiite-majority Islamic republic, he said, stressing that their alliances with Washington would remain unchanged.
"Iran still poses challenges to our interests and values," the US leader told reporters, citing "its support of terrorism and its use of proxies to destabilise parts of the Middle East."
Rouhani "must be concerned about possible violation of commitments by the other parties and close paths to it," the leader wrote in the letter, published on his website.
"You are well aware that some of the six states participating in negotiations are not trustworthy at all," Khamenei said.