(Reuters) - U.S. President Barack Obama will visit Saudi Arabia on Tuesday to meet its new King Salman after the death of his predecessor Abdullah on Friday, testimony to Riyadh's important role in energy markets and the fight against Islamist militancy.

Russia's Prime Minister Dmitry Medvedev, Britain's Prime Minister David Cameron, French President Francois Hollande and Japan's Crown Prince Naruhito will visit on Saturday and Sunday to offer condolences.

Obama will cut short a visit to India to pay his respects instead of Vice President Joe Biden, who had been scheduled to fly in from Washington, the White House said on Saturday.

Muslim leaders paid their respects on Friday at Abdullah's funeral in Riyadh.

Some international human rights groups, meanwhile, have called on Western leaders to condemn Saudi Arabia's record of crushing dissent and depriving women and foreign workers of rights instead of praising its late king.

Salman takes charge in Saudi Arabia at a time of deep uncertainty in the kingdom, surrounded by a region in tumult and nervous about both Iranian influence and the spread of Islamist militants.

The kingdom's role in orchestrating Arab support for joint action with Western countries against the Islamic State group has won praise in Washington, while its role as biggest oil exporter is particularly important at this time of market instability.

Saudi jets have bombed IS targets in Syria, its top clerics have issued repeated denunciations of the militant group despite similarities between its ideology and Wahhabism, and the police have detained thousands of militant suspects in the past decade.

Salman pledged on Friday to maintain the kingdom's policies and kept most of Abdullah's cabinet, including the oil, finance and foreign affairs ministers, in place.

Western countries also value the kingdom as an important market for their defense industries and Salman quickly moved to appoint his son Prince Mohammed, 35, as his own successor as Defence Minister, responsible for big arms contracts.

Saudi Arabia observes no official period of mourning, in keeping with the ascetic traditions of its official Wahhabi school of Sunni Islam, but the royal court has announced that it will receive condolences and pledges of allegiance until Sunday.

Late on Friday state television showed princes, Wahhabi clerics, tribal chiefs, military leaders, major businessmen and other dignitaries crowding the royal palace to kiss King Salman's shoulder or hand.

The king's rapid appointments of half-brother Muqrin, 69, as Crown Prince and nephew Mohammed bin Nayef, 55, as Deputy Crown Prince, appeared to resolve for many years to come speculation that succession disputes might destabilize the ruling family.

(Reporting By Angus McDowall Editing by Jeremy Gaunt)