"We have been stormed in the past six months with painful and negative developments... but we managed to achieve calm," Sheikh Sabah said as he opened the Gulf Cooperation Council (GCC) summit.
Sheikh Sabah has been mediating to try to resolve a bitter dispute between Qatar on one side and Saudi Arabia, the United Arab Emirates, Bahrain and Egypt on the other.
On June 5, the Saudi alliance cut off all diplomatic and economic ties with Qatar, accusing it of links to extremist groups. Doha has denied the allegations.
Saudi King Salman and the leaders of the UAE and Bahrain are absent from the summit, but Qatar's Emir Sheikh Tamim bin Hamad Al-Thani is attending.
"We will continue this role (mediation) in the dispute," the Kuwaiti emir said. "Our meeting today is a reason to continue the mediation which fulfils the ambitions of our people."
Sheikh Sabah also called for the GCC charter to be amended to establish a clear mechanism for resolving disputes between member states.
Founded in 1981, the GCC groups Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and the UAE.