“The Appearance of the Horseshoe Arch in Western Europe,” by Ernest T. Dewald, American Journal of Archaeology, Vol. XXVI (1922), No. 3, pp. 316-331.
It’s a late Roman thing, found throughout both the Western and Eastern Empire. The Muslim invaders of the Eastern Empire took it over, and brought it along to Spain with them; but there were already plenty of horseshoe arches in Visigoth Spain when the Muslims got there. They had been there for centuries. A lot of Roman Syrians had emigrated to Spain in various ways and for various reasons, and they brought their architecture along with them.
The earliest known example of the horseshoe arch in Syria dates from the 2nd century A.D. In some ancient Syrian churches, the entire apse is a horseshoe arch shape. There is a horseshoe arch in Pompeii, and another in the Pantheon. The church of St. Apollinare in Ravenna has a horseshoe arch-shaped apse.
Dewald also points out that on the Canon Tables pages of Gospel manuscripts (which usually portray arched pillars “roofing over” the comparisons of the Gospels), ancient Syriac illuminations show horseshoe arches. (As do many Irish mss.) This compares directly to the use of horseshoe arches in Beatus mss.
Dewald also has some interesting things to say about the Mozarabic Rite really being influenced strongly by the Syriac Fathers, via all those Syriac immigrants mentioned before. So it’s worth a read.