Gov. Ron DeSantis of Florida referred to the Gulf of Mexico as the “Gulf of America” in a state executive order on Monday, even before President Trump had taken any presidential action to rename the body of water.

Mr. Trump said during his inaugural address that one of his first actions in the White House would be to rename the Gulf of Mexico as the Gulf of America, but he had yet to take action on the promise hours into his second term. The pledge during his inaugural speech drew an audible laugh in the Capitol Rotunda from Hillary Clinton, the former secretary of state and two-time Democratic presidential candidate.

But it was no laughing matter for Mr. DeSantis, who attended the inauguration on Monday and sat in the overflow room in Emancipation Hall, where he watched the proceedings on a video screen with other Republican governors. He didn’t wait for Mr. Trump to take any presidential action before forging ahead on his own in an early act of self-implementation that appeared indicative of how Republicans seeking to please Mr. Trump may act as he outlines his agenda.

In an executive order describing a brewing winter storm heading for the Florida Panhandle, Mr. DeSantis catered to Mr. Trump in his official language as he declared a state of emergency in his state.