Courageous President, resolute people
Iran's government is an Islamic theocracy in which the highest official is not the President, despite his title, but rather the Supreme Leader, a religious office currently filled by Ayatollah Ali Khamenei. The President does have real power and is elected by the people, but the religious establishment is not above banning candidates it doesn't like and rigging elections, as in the case of 2009 when Rouhani's predecessor, President Mahmoud Ahmadinejad (a standard-issue Israel-hating Islamic religious nutjob) was "re-elected", triggering the Green Movement, mass street protests which were among the largest in world history. It was these protests which intimidated the theocrats enough that they refrained from blocking Rouhani's election in 2013, despite their hostility to his reformist tendencies.
Most of the machinery of internal repression in Iran is outside the President's official jurisdiction, but as in the US, a popular President can exert influence even on issues outside the formal scope of his office.
Lately Rouhani has begun doing this. In May, for example, the regime imposed tighter hijab rules (there is a constant tension in Iran between women pushing the limits of the Islamic dress code and theocrats trying to enforce it more strictly), and several young people were arrested for making a dance video. As public exasperation over these repressive measures mounted, Rouhani spoke out:
"Do not interfere in people’s lives so much, even if it is out of compassion," Rouhani said at a health insurance conference on May 24 arranged by the administration. "Let people pick their own path to heaven. One cannot take people to heaven through force and a whip. The Prophet [Mohammad] did not have a whip in his hand."
Religious hard-liners pushed back hard in Friday prayer speeches in mosques across the country, which are a regular venue for making the theocracy's official positions on issues known. But rather than backing down, Rouhani made further criticisms of, for example, internet censorship (see again the link which has much more detail).
A couple of weeks later, during public protests calling for the release of political prisoners, Rouhani pithily twisted the theocrats' tails again (details here) and his Minister of Culture ridiculed the way authoritarians' suspicion of new communications technologies leads to continual backwardness (a problem which contributed to the fall of the USSR). Again the conservatives reacted with alarm, denouncing Rouhani's view that "we don't take people to Paradise by force" as "liberal and modernist", which are apparently very bad things.
Rouhani's cabinet appointees are another example of his efforts to reform Iran domestically. See for example his Minister of Science (who has authority over education and research), whose liberalizing actions have provoked conservatives to try to impeach him.
But the most startling confrontation has arisen over an issue that is also a liberal-vs-theocrat battleground in the US -- contraception. Surprisingly, Iran for many years has had government-guaranteed access to birth control (scroll down about halfway), even in remote villages which are visited by traveling gynecologists for the purpose. But last month Supreme Leader Khamenei announced a change of course, supporting a proposed law banning many forms of birth control, and declaring a goal to increase the population from its current 77 million to 150 million.
Rouhani has now denounced the proposed law as a potential violation of human rights and declared that his administration will oppose it -- despite that fact that this means directly opposing the will of the Supreme Leader. It takes real courage to stand firmly for free public access to birth control in a political system whose Scalias and Alitos are nominally his superiors in authority, and have armies of thugs at their command as well as a proven willingness to use torture and murder to crush their enemies.
So there is no longer any doubt -- Rouhani is indeed trying to reform Iran domestically despite great obstacles and risks. And the key to his ability to do so is popular support. Remember, the Ayatollahs aren't afraid of the United States. They're afraid of a recurrence of this:
Remember, Iran is potentially a country of enormous influence. Historically its culture and language have great prestige in the Middle East. Even under this wretched theocracy it is one of the most advanced countries in the area -- it has a convincing nuclear-weapons program and a space program, it conducts stem-cell research, it manufactures cars, it teaches evolution in its schools without reservation (putting it ahead of places like Louisiana), it guarantees access to birth control (until and unless Khamenei gets his way). Only de-fang the theocrats and lift their internal repression, and it would probably advance as rapidly as Spain and Portugal did after the end of fascism there.
I haven't yet heard of Rouhani taking any action on the murderous repression of gays or the persecution of the Baha'is, but perhaps that will come next. He has a lot of targets to choose from.
[The majority of my information on this comes via Iranian blogger Kaveh Mousavi, who provides a very valuable insider's view of what's happening in Iran, from a secular perspective.]