"Foreign forces can cause problems and insecurity for our people and for our region", Rouhani said in a televised speech at an annual military parade, adding that Iran would present to the United Nations a regional cooperation plan for peace.
Britain had previously held back from attributing blame for the drone and missile attack.
Iran denied involvement in the attack.
In a speech at Monday's parade, Rouhani denounced the United States presence.
USA has slapped more sanctions on Iran, penalizing the Iranian Central Bank while the Pentagon said it was sending US troops to bolster Saudi Arabia's air and missile defenses after the largest-ever attack on the kingdom's oil facilities.
The Pentagon announced on Friday that, in response to a Saudi request, there would be a moderate deployment to Saudi Arabia, not in the thousands and focused on air and missile defence.
Iran's president on Sunday urged Western powers to leave the security of the Persian Gulf to regional nations led by Tehran.
President Rouhani is expected to travel to NY on Monday, a day before general debate kicks off at the United Nations general assembly.
Tensions between the two countries have been on edge since Trump previous year pulled the USA out of the 2015 nuclear deal between Iran and world powers and reimposed sanctions against the Islamic Republic.
Johnson said he would meet Rouhani at this week's high-level United Nations gathering.
He said on his way to NY: 'I can tell you that the United Kingdom is attributing responsibility with a very high degree of probability to Iran for the Aramco attacks.
Asked if military action was possible, he replied: 'We will consider in what way we could be useful if asked and depending on what the exact plan is'.
Iranian Foreign Minister Mohammad Javad Zarif, who is in NY for the U.N. meetings, has warned that any retaliatory strike on Iran by the United States or Saudi Arabia will result in "an all-out war".
"The fund community faded the attack last week on the assumption that supply would return very quickly, but the reality is likely to be different", said Energy Aspects analyst Virendra Chauhan in Singapore.
France has led a European effort to try and defuse tensions between Washington and Tehran but those efforts have stalled, with Iran reducing its commitments to a 2015 nuclear deal with world powers and the United States refusing to ease sanctions that have strangled Iran's economy.
While Yemen's Iranian-allied Houthi rebels claimed the assault, Saudi Arabia says it was "unquestionably sponsored by Iran".
"This means the U.S.is completely desperate", he said, suggesting the USA had little leverage left over Iran.