"We had to cancel because of the flight situation... and the ongoing fighting in the region," a Sri Lanka Cricket official told AFP.
The teams were scheduled to play three T20 internationals in Sharjah on March 13, 15 and 17, and three one-day internationals in Dubai on March 20, 22 and 25.
It would have been the first time that Afghanistan had hosted Sri Lanka for a bilateral series.
Fighting has spread across the Middle East since the United States and Israel attacked Iran on February 28, with Tehran launching retaliatory strikes.
The conflict has disrupted flights across the region, including in Dubai, where the airport was briefly closed on Saturday as Iran fired drones and missiles at targets across the Gulf.
War-ravaged Afghanistan has never hosted an international cricket match, instead having to play its home fixtures in India or the UAE.
