It would be a good deal more accurate to say the hunting down and killing of priests and nuns was primarily a consequence of the rightist coup, not its cause.
<50 priests were killed before the war started, over a several year period. During the war, and especially in the first few months, something over 5000 clergy were murdered.
To be fair, the other side was doing pretty much the same thing, killing members of groups it saw as opponents, such as union members and leaders, schoolteachers, etc.
To be even more fair, both sides were "right" to do so in a sense. Most of the members of the clergy were opponents of the government, and most schoolteachers were opposed to the rightists.
The Red Terror death toll has been estimated at 40k to 110k. The White Terror killed somewhere between 200k and 400k, but then they won, and they had a lot longer to do their killing in.
Of course, not all of those killed by either side were innocent victims.
The church- and convent-burnings were started before the war (they convinced the generals they had to fight); the leftist government outlawed the Jesuits as well. Read “The Last Crusade” and you’ll understand how it happened; understanding the Spanish Civil War is key to understanding many Europeans’ tolerance/support of fascism (as opposed to Stalinism, which they witnessed in Spain and of which they were terrified). The arrest of the Falange leader months before the war was important as well; imagine Obama’s first executive order was to arrest Mitt Romney.
There is no comparison between the wrongs committed by the generals and the communists in Spain between 1936 and 1939; in 1937 the communists had murdered many of their own allies (the anarchists) in their own civil war. Once they were exposed for what they were, the “Republic” could never garner enough public support to win. In fact, they steadily lost ground from the first days of the war, and murdered as many people as possible until the army (assisted by Hitler & Mussolini) stopped them.
“The Red Terror death toll has been estimated at 40k to 110k.”
40K is an absurd figure when you realize they have the names of 7K priests alone that were murdered.