Less than a year after Iranian President Hasan Rouhani was elected in a landslide vote on a platform of change, he faces his first domestic crisis, as public anger roils his administration.

Many Iranians who voted for Mr. Rouhani in the expectation that he would improve human rights and the economy have protested his government in recent weeks over what they say is his neglect of domestic issues, as he focuses on foreign policy.

On...