Strictly speaking, there was no theological justification for the concept of limpieza de sangre. The roots of this are in discrimination against the converso and morisco communities, who were converts from Judaism and Islam respectively.
There had always been a level of enmity and competition between the different religious groups in Spain. And some vocations were open to people of certain religion but not to others. As an example, in most constituent states of Spain, only Christians could hold public office. So Jews tended to work as traders, tax gatherers, money lenders, etc. Christian Spain had previously been seen as very liberal in their treatment of Jews, because the Jews were allowed to own property, unlike in most other Christian states of Europe!
However, as some Jews converted, they became eligible and in some areas such as Burgos and Toledo, they became a strong presence in the city council, to the ire of the self-identified Old Christians who viewed them as having their cake (continuing to work as money lenders, tax gatherers, etc.) and eating them too (by becoming councilors).
The end of the Reconquista in 1492 heralded a new policy by Isabella and Ferdinand, where they increasingly asserted stronger and stronger pressure to Jews and Muslims to convert to Christianity. Some were genuine, some less so. Some maintained their cultural heritage more strongly than others. Regardless, even though Ferdinand II Aragon was openly identified as coming from converso origins, by the time of Charles V there was a movement by Old Christians to start to associate descent from Jews or Muslims as a means of discrimination in what was seen as competition. This increased under Philip II, when Cardinal Mendoza y Bobadilla published a pamphlet denouncing the nobility as having been "infiltrated" by Jewish blood. The Cardinal was indeed an angry man, as several of his family members had been denied public office, so in revenge he wrote the pamphlet literally to stir up trouble. Purity of blood became the basis of status struggle.
Further, as the Spanish Inquisition grew, they did not use purity of blood (or lack of) as evidence, however it was often used in support of whether to launch an investigation to offenses which were officially considered admissible. So this was convenient for Old Christians who held grudges against Converso rivals, as they could helpfully suggest that the Inquisition investigate such and such family because they are impure, and thus likely to be crypto-Jews, or even worse, trying to convert old Christians into Jewish faith.
TL;DR Purity of blood became the basis of status struggle. There was no theological basis for it.