President Ram Nath Kovind administered the oath to Justice Gogoi at a ceremony which took place in Rashtrapati Bhavan's Darbar Hall.
Justice Dipak Misra served as the 45th Chief Justice of India, from August 28, 2017, to October 2, 2018.
The bench said: "We find that it is not a case we need to interfere. We do not find merit in this. Your writ of quo warranto is devoid of merit."
Justice Ranjan Gogoi will have a little over one year before he demits office on attaining the age of 65. He will retire on November 17, 2019.
Justice Ranjan Gogoi will be the next Chief Justice of India. He will take over from Chief Justice Dipak Misra, who is retiring on 2 October.