How could it have caused a war?
Yes, it could have caused a war. Indeed, it came close to doing so in OTL.
One interesting thing is that you would probably have had Phil Sheridan commanding the army in Cuba, and very possibly Nathan Bedford Forrest commanding the cavalry (he wrote to Sherman during the crisis requesting such a command, and Sherman sent the letter on the War Department with favorable comments, while writing back to Forrest and telling him that he thought War with Spain was unlikely, and therefore the Army could not offer him a commission at that time).
Another interesting thing is that the U.S. Navy at that time was totally outclassed by the Spaniards. The Spanish had a modern navy, with a good number of ironclad warships. The U.S. had scrapped all its ironclads after the Civil War and the only ships it had under construction in 1873 were wooden screw sloops similar to designs dating back to the 1850s. So any invasion of Cuba would have to have waited until the U.S. could build some ocean-going ironclads to escort an invasion fleet.