Abd al-Rahman (the Umayyad who escaped the Abbasid uprising) wasn't the Caliph at the time; he was only a prince, so he wasn't actually demoted. He was pursued across North Africa by Abbasid assassins, before he reached the safety of his Berber mother's tribe. Al-Andalus (Iberia) had experiencing political turmoil between various factions for several decades, so he raised an army to back one of the factions and took control for himself, using his Umayyad lineage to legitimate his own rule as Emir Abd al-Rahman I.

While Abd al-Rahman had no hesitation about using violence when necessary, his first order of business was to get the Abbasids to call off their dogs. al-Andalus needed to be put back together after a long period of turmoil, and he didn't have the resources to take on the Abbasids, either in Spain or on their own turf.

When he discovered Abbasid spies at court, he had them beheaded and the heads sent to the new caliph, who is said to have declared "God be praised for placing a sea between us." The caliph basically decided to back off; although there was no formal truce, the understanding was that the Abbasids wouldn't interfere in Spain, and the Umayyads would stay there.

Abd al-Rahman declaring himself caliph would have broken the truce by directly challenging the Abbasid right to govern. Instead, he started the steps toward making al-Andalus a rival in the west. When his descendant, Abd al-Rahman III eventually declared himself caliph, the Abbasids were overextended and weak, but even then it was more of a declaration of independence rather than a declaration of war.

See: Chris Lowney, A Vanished World: Muslims, Christians, and Jews in Medieval Spain, 64-66; Antonio Muñoz Molina, Córdoba de los omeyas, Chapter III: El Príncipe Fugitivo.