Issa said on Fox News' "America's Newsroom" that it was unlikely Congress would be able to contest the closure in a timely fashion.
"Our process is to go to the court. The court is likely not to rule quickly," he said. "It's an old example, but Andy Jackson, the founder of the Democratic Party, in the Trail of Tears defied the Supreme Court and marched Native Americans to their deaths."
The Trail of Tears was the forced relocation in the 1830s of the Cherokee Nation to Oklahoma under former President Andrew Jackson's Indian Removal Act. The Cherokee Nation says roughly 4,000 people died.
Issa said it was up to U.S. military leaders to decide whether to follow Obama's plan, which he said was unlawful.
"Are they going to obey an unlawful order, an unlawful order to move people from Guantanamo, an unlawful order to close the base?" he said.
The California congressman added that the deaths of thousands of Cherokees occurred because the military followed a President's order rather than the law. He further suggested he'd support military leaders who refused to implement Obama's plan.
"Remember the Trail of Tears was only possible, the murder of those Native Americans was only possible, because the military obeyed an order in violation of the U.S. Supreme Court," he added. "So do I believe that the military may push back on the President if he's given an unlawful order? Actually, I do. I can see flag officers resigning rather than obeying a clearly unlawful order."