Once the Germans had decided on unrestricted submarine warfare (including killing Americans on American ships [1]) it was inevitable that the US would enter the war. (At least under President Wilson or for that matter President Hughes--if there were a President Bryan or a President La Follette, it might be different.) The Zimmerman Telegram may have made the vote for war in Congress even more overwhelming than it otherwise would have been, but that is all.
In fact, the whole reason for the Telegram is that the Germans realized that their decision for unlimited submarine warfare was almost certain to lead to war with the US, and wanted the support of Mexico in the event that occurred.
[1] Rather oddly, many people here don't seem to get the significance of this. Prior to 1917, there had only been one incident of Germans killing Americans
on an American ship--the Gulflight incident.
https://en.wikipedia.org/wiki/Gulflight And there "As an American ship, the submarine would not have attacked had he seen her nationality, but apart from an ordinary flag
Gulflight was not carrying any additional markings painted on the hull to make clear her nationality, which other ships were then doing." A lot of Americans thought that Americans had no business traveling on belligerent ships but drew the line where attacks on American ships were concerned. Now in 1917 fort he first time Germany had made sinking such ships a
policy.