Both sides in the Spanish civil war were very diverse with often conflicting goals and aims.

There were certainly fascists within the nationalist side - the falange were a fascist organisation and it was part of the grand coalition on the nationalist side. However, the nationalist side included royalists and monarchists (both Bourbonists and Carlists), centralists (which feared the local autonomy given to regions of Spain), conservatives, reactionaries, dissatisfied military, religeous fanatics and many others.

The most common trait of the various nationalist factions were anti-communism and anti-anarchism.

The Falange was a bit too radical at times for the traditionalists in other groups, and after conflicts within the nationalist coalition, it was force-merged into the grand nationalist party and effectively neutered - despite getting considerable support from Italy. About 60% of the original members of the Falange died during the civil war, and Franco further weakened their numbers by forming the Blue Division of Spanish volunteers that fought on the Eastern Front with the Germans 1941-1943. Many of the remaining original members of the Falange joined this force.

So, the nationalists included one purely fascist organisation. Many of the other parts and the grand nationalist party (or bloc) formed from merged organisations also showed fascistic tendencies - violent anti-communism and anti-anarchism, glorification of the army and of violence, however, most of them had other traits such as conservatism, religion and royalism which are not fascist traits.

One thing they shouldn't be called, even if everyone in English does, is Nationalists. That's a mistranslation of the name they used themselves, Nacionales ("Nationals"). The Spanish equivalent of "Nationalists" is Nacionalistas, and at least in modern Spanish political discourse, is nearly exclusively used to refer to Basque and Catalan nationalists, separatists or not. These fought in the war against the Nacionales.

The answer from /u/vonadier covers most of the major points. The divisions within the Nationalists existed, but Franco's major strength (in my opinion) was his ability to subsume the various factions within the Nationalists within and under his own banner. The monarchists were often at each other's throats before the war broke out, and both happily took pot shots at the Falange (Spain's fascist party movement). All three were derisive of the CEDA (a Catholic centrist party). Militarists and traditionalists were at odds with any attempt to change how things were run, and some reform measures came from the right after 1931. But Franco was able to unite these disparate groups and keep them fighting the Republicans rather than each other. This was helped by the leader of the Falange being in Republican custody when the war started, as well as the deaths of two competing generals. Other faction leaders willingly moved aside, and others were marginalized.

That still doesn't answer whether we should call the Nationalists "fascists." The Falange was a pretty small group before the Spanish Civil War began. They got less than 0.1% of the electoral results in the 1936 elections and received less than 7,000 votes out of the 9,450,000 cast. That's tiny. Now, some supporters of fascism would have voted for other rightist parties so it's not conclusive, but it's a clear indication that the Falange was not as powerful as the Nazi party or Mussolini's National Fascist Party. Its membership roles skyrocketed during the war (and Paul Preston covers this very well in The Spanish Holocaust). Some of these new members were committed fascists, others joined out of a desire for self-preservation (I'm not an anarchist, I'm a Falangist now). Still more were swept up in the fervor of war. Throughout the war, Franco was able to harness this growth of the Falange while undermining Falangist leadership (especially in that the Falange's leader was imprisoned and later killed by the Republicans). Franco did the same to military leaders (and again was helped by two separate plane crashes killing two prominent generals that could have challenged him for position), religious leaders, monarchist proponents, and industrialists and large landowners. No one group of these people was allowed to dominate the others. Franco played these groups off of each other but kept them from outright violence and was thereby able to gain control over the entire country.

So, no, I wouldn't call the Nationalists "fascist." That's not exculpatory either--the Nationalists killed roughly 150,000 noncombatants during the war in a deliberate attempt to destroy leftist movements in Spain (the Republicans killed roughly 50,000 noncombatants). Jose Sanchez's characterization of Franco's regime is apt: he calls it "barbarous." The Nationalists can be fairly characterized as traditionalists, reactionaries, and nationalists--just not fascists.