The demand came a day after the prosecutor for the state of Michoacan announced police had found Adame's charred body -- more than a month after he was kidnapped by gunmen -- and were investigating a "personal" dispute with a regional crime boss as the motive.
Adame, the owner of local TV station Canal 6, was the sixth journalist killed this year in Mexico, which is ranked the third-most dangerous country in the world for reporters after the war zones of Syria and Afghanistan.
The suspect, Daniel "El Cabezas" Rubio, told investigators that Adame -- whom he described as a close friend -- had been killed on the orders of a regional crime boss known as "El Chano Pena."
Godoy said evidence suggested the crime boss and Adame had "problems of a personal nature" -- possibly a love triangle.
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But the journalist's family and colleagues questioned that version of events at a press conference.
"We don't know this man," his daughter, Frida, said of "El Cabezas."
Fellow journalists in Michoacan -- a western state hit hard by Mexico's epidemic of gang violence -- urged prosecutors to examine that as a possible motive.
"We demand the Michoacan government exhaust all lines of investigation, particularly those related to Salvador Adame's journalism," said his colleague Patricia Monreal, representing the state journalists' union.
Adame, 44, was kidnapped a day after President Enrique Pena Nieto had vowed to strengthen protections for journalists and prosecute those who attack them.
Since then, at least 100 journalists have been killed and more than 20 have disappeared.
The violence is part of a wave of bloodshed in Mexico over the same period that has left more than 2,00,000 people dead or missing.