I doubt he would join Teddy, he was fairly old by the time of the Spanish-American War. After Tombstone, he and Josie were committed to each other and their adventures. I could see him running for mayor of a boomtown in the Yukon, or perhaps a member of the California Assembly when he retires out there.
Earp is only 8 years older than Theodore Roosevelt/.
That said he is a lawgiver, and does not seem soldierly imo. But for some reason i would now like to see Vice President Earp.