From The Trident Society's The Book of Navy Songs (1926):

"Col. Henry C. Davis, U. S. Marine Corps [1877-1929], writes as follows:
"'I have never been able to trace the origins of the song beyond the words of the first two lines,'"From the halls of Montezuma to the shores of Tripoli,"' which was inscribed on the corps colors from a date of many years ago.

"The following verse I wrote at Camp Meyer [at Guantanamo Bay] in 1911 when on an expedition."

The Book of Navy Songs gives three verses, and from the presentation it isn't clear which one Davis wrote, or whether he wrote both verse 2 and verse 3.

Verse 2 is the one beginning, "Our flag's unfurl'd to every breeze," and verse 3 begins, "Here's health to you and to our corps."