Ms Bishop’s decision came after she was knocked out in the first round of the three-way Liberal leadership contest which installed Mr Morrison as PM in place of Malcolm Turnbull.
Mr Turnbull described Ms Bishop as an inspiring role model for women “here and around the world”.
“Today we have lost Australia’s finest foreign minister,” he tweeted.
Labor’s Foreign Affairs Spokeswoman Penny Wong paid tribute to Julie Bishop “for her trailblazing role as the first Australian woman” to be foreign affairs minister.
“For five years she has dedicated her life to our nation with a tireless work ethic and exhausting travel schedule,” Senator Wong said.
She said Labor had at times been critical of foreign policy directions under prime ministers Abbott and Turnbull but “Ms Bishop’s commitment to standing up for Australia both here and abroad has never been in question”.
“In particular I have deeply appreciated her commitment to bipartisanship and her personal courtesy to me,” Senator Wong said.
Up until Friday’s leadership spill Ms Bishop had been deputy leader of the Liberal Party for more than a decade, having taken on the position in the aftermath of the Howard Government’s election defeat in November 2007.
She had served as foreign affairs minister since the Liberals regained government under Tony Abbott in September 2013.
As foreign minister, Ms Bishop was tasked with how to deal with an expansive China.
She also campaigned in the United Nations to get access to the Malaysian Airlines flight MH17 crash site after the downing of the plane over rebel-controlled eastern Ukraine in July 2014.
After a joint investigation into the incident concluded the Russian military was behind the missile attack, she demanded the suspects be extradited to the Netherlands to face trial.