There were certainly international aspects to the Spanish Civil War. There are some scholars who have used the term "proxy war," including Joseph Maiolo in Cry Havoc. Certainly the fact that Germany and Italy backed the Nationalists while the Soviet Union backed the Republicans gave the war the feeling of fascists vs. communists in a warmup for the coming general European war that most perceived to be on the horizon. This perception persisted into WWII, with some hoping that antifascism would include a drive to remove Franco once Hitler and Mussolini were dealt with.

However, this narrative doesn't stand up to much scrutiny. Most of the combatants on both sides were Spanish. The conflict that broke out was not precipitated by outside actors. Many on both sides welcomed an armed conflict because they were certain that their faction would emerge victorious--especially among the Nationalists. The aid from Germany and the Soviet Union to the two sides was roughly equal, while the largest foreign contingent from one nation was the Italian CTV (which had spotty performance during the war). The biggest advantage given to either side was the German and Italian states loaning aircraft to the Nationalists in the opening weeks of the war so they could airlift the Army of Africa to the Spanish mainland. France and the UK were committed to the strategy of nonintervention (and Germany, Italy, and the Soviet Union were also technically on board) so even the biggest European powers were not all involved.

Even characterizing the conflict as being between fascism and communism oversimplifies the situation. The Nationalists were made up of fascists (the falange), much of the military hierarchy, two different factions of monarchists, the Catholic CEDA political party (with a distressing amount of the Catholic hierarchy endorsing the rebellion), industrialists, large landowners, and other traditional elites. The Republicans were made up of communists that were divided into two major factions, numerous socialist groups, even more numerous anarchist associations, various unionized labor groups, some people dedicated to republican government, regionalists like the Basques and Catalans, and many other groups. The Spanish Civil War was far more complex than merely Nationalists vs. Republicans, much less fascists vs. communists.

So while there was indeed a great deal of international involvement and interest in the Spanish Civil War and a number of nations chose their policies regarding that war based on their own interests, the Spanish Civil War was a local conflict at its heart. In no way should that diminish the impact of other nations on the war, but that argument can be easily overstated as well.

As always, followup questions from OP and others are encouraged!