Ecuador have now gone seven games without a win but Gomez was not for stepping down.
"None of the officials hired me saying what happens to me depends on what I do in the Copa America," he told reporters.
"Because if they had then I wouldn’t have taken the job.
"I’m not handing over my post. If they want to fire them, then fire me.
"Another coach will come in and the same things will happen, they are asking for results when we can’t provide them," he said. "Let’s see who explodes first.
"I don’t think I am the problem."
Colombian Gomez took Ecuador to their first World Cup finals in 2002 during his first spell in charge. He returned to the post last year after Ecuador fell apart following a brilliant start to the South American qualifiers for Russia 2018.
Four wins in their first four games were followed by just two more in their next 14, leaving them well outside the qualification places and costing Patricio Lara his job.
Gomez won four of his first six games in charge but they have toiled since.
(Reporting by Andrew Downie; Editing by Peter Rutherford)
