Medvedev also ordered the Interior Ministry to recommend transport security officials for dismissal and said authorities found culpable would be held responsible.
Both Putin and Medvedev donned white coats and stood at victims' bedsides in separate visits shown on state television.
Putin, the dominant partner in Russia's 'tandem' leadership, built his tough reputation by launching a war in late 1999 to crush a rebel government in Chechnya, a North Caucasus province.
That campaign achieved its immediate aim but insurgency has spread to neighboring Ingushetia and Dagestan and spawned persistent attacks beyond the North Caucasus, despite Kremlin vows to crush insurgents and nurture the region with subsidies.
Government critics warned that tough rhetoric will do little to stop attacks by militants in an insurgency they say is aggravated by heavy-handed law enforcement.
The choice of Domodedovo international arrivals area suggested the attackers wanted to make an impact beyond Russia.
An investigator cited by news agency Itar-Tass said the bomber appeared to have been a heavily built man aged 30 to 40. Other reports pointed to a female bomber or two attackers.
An Emergencies Ministry list of the dead included eight foreigners: two Britons, a German and citizens of Bulgaria, Kyrgyzstan, Tajikistan, Uzbekistan and Ukraine. Colleagues said the Ukrainian was Anna Yablonskaya, 29, a playwright who was traveling to Moscow to receive an award.
Britains Foreign Office said one Briton was confirmed dead.
The spread of violence from the North Caucasus, where it is fed by corruption, poverty, clan rivalries and religious radicalism, fans Russian nationalist militancy in the heartland.
Tensions between ethnic Russians and the 20 million Muslims who make up one-seventh of Russia's population flared dramatically last month when Russian nationalists attacked passersby of non-Slavic appearance, many of them from the North Caucasus, in central Moscow -- just steps from the Kremlin.
On Tuesday police officers boosted their presence around railway stations and airports, carrying out spot checks of people who looked as though they could be from the Caucasus.