In his inaugural speech, Rouhani told parliament: "All of those who voted, whether they voted for me, someone else, or even if they didn't vote, all of them are Iranian citizens and have citizenship rights."
Rouhani, who replaced controversial President Mahmoud Ahmadinejad, said the people had voted "yes" to moderation and hope. He promised to advance women's rights and freedoms and to reduce the government's interference in people's lives.
On many issues, whether Rouhani, a former nuclear negotiator, can deliver will depend on his relationship with the Supreme Leader Ayatollah Ali Khamenei, the BBC reported.
Khamenei is effectively the leader of the Islamic hardliners, and has the last say on many crucial and strategic issues.
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While Rouhani needs the hardliners to co-operate, the hardliners need Rouhani to save the regime from the deep trouble it finds itself in as international sanctions and mismanagement of the economy erode its authority at home.
On the economic front, with a negative growth, the Iranian economy is in recession.
Inflation for food prices has touched 55 per cent. Unemployment is around 12 per cent and rising. The mismanagement has been monumental, but international sanctions have wreaked havoc too. Sanctions on Iran's oil exports have reduced its main source of income by about 65 per cent, the report said.
Banking sanctions have had an even more disruptive impact on Iran's trade with the outside world, making it impossible for Iran to bring its petrodollars back into the country - hence the shortage of hard currency that has led to a huge drop in the value of the Iranian rial, by about 80 per cent in the last year.
Rouhani has earlier said that it is possible for Iran to maintain its nuclear programme and at the same time reassure world powers. He has suggested that Iran may be more proactive in the talks, and more transparent in its nuclear activities. He said he wants to arrive at a position of mutual trust with world powers.
In his first public comments after assuming power, Rouhani said, "The only path to interact with Iran is through negotiations on equal grounds, reciprocal trust-building, mutual respect and reducing hostilities".
"All this are easier said than done. But if he finds a way, he may well have the backing of Iran's Supreme Leader. If not, the spectre of war will continue to hang over Iran," the BBC said.