There's a lot to unpack in that question, and quite a bit of it lies beyond what I'm comfortable discussing, but I'll try and address the parts of it that I can. There are a couple of misconceptions and myths underpinning your question which I'll address, and hopefully in doing so at least partially answer your question. First off, let's lightning round through some basic fact stuff before getting to the meat of the answer:

1. The Arab Conquests took place in the Early Middle Ages. They're actually a series of campaigns and conquests wherein Arab rulers conquered huge swaths of territory over the 7th and 8th centuries, including but not limited to all of Northern Africa.
2. The conquest of Iberia (the term for the collective peninsula including both modern day Spain and Portugal) took place toward the end of the Arab Conquests and was a combined effort of Arabic rulers who had swept across North Africa and the native Berber peoples (note: there is some debate about the legitimacy of using the term "Berber", as it has the same derivation as "Barbarian" and means largely the same thing...so it's not the best label and not always applied with proper exactness, but it's what most historians in my period use so we're going with it). These are two distinct groups, Arabs from the Arabian Peninsula (think modern day Saudi Arabia and surroundings) and natives from Northwestern Africa and the Sahara (Morocco and its neighbours today).

Now the big one, and it's a pretty common misconception: The Arab Conquests didn't involve wholesale sweeping away of the native populations and replacing them with a new Muslim Arabic population. You most often encounter this myth when discussing places that are currently majority Arabic but weren't always, e.g. Egypt. In the case of Egypt, the Arabic Conquest involved the removal of the previous ruling dynasties (mostly remnants of Byzantine rule) and replacing of them with Arabic Muslim rulers. In practice, the population remained relatively unchanged and as late as the 12th century the population was still majority Christian (mostly Coptic and Orthodox). Now, Egypt was very close to the Arabian homeland and would have seen even greater resettlement than somewhere as far flung as Iberia.

There would have been some resettlement, obviously the rulers and their families came but they also brought Berber mercenaries with them to enforce their rule. However, there was a lot of mistrust between the Berber armies and the Arabic rulers - despite both being Muslim they weren't always culturally compatible - so Arabic rulers tried to limit the number of Berber soldiers in Iberia (or al Andalus as they called it) and kept the forces over in Africa as much as possible, at least initially.

To greatly abridge a vast and fascinating history, Muslim Spain had four major periods: the initial post-conquest period that led to the rise of the Umayyad Caliphate (Arabic rulers under the ostensible control of a religious leader), the Taifa Kingdoms (multiple small kingdoms in near constant conflict in the semi-anarchy after the fall of the Caliphate), the Almoravids (Berber rulers who conquered the Taifa kingdoms from a power base in North Africa) and the Almohads (a different group of Berbers who conquered the Almoravids from a power base in a different part of North Africa). Now, Muslim rulers continued to exist in Spain for a long time after the collapse of Almohad authority in the 1200s, but they controlled only a tiny section on the southern edge of the peninsula and the rest of Iberia was under strict Christian rule.

I highlighted the above dynasties to show the waves of conquest and migration that happened across the peninsula, and it's worth noting that a very similar thing happened on the Christian side. Christian nobility from across Europe (but especially Burgundy in France) married into Spanish noble families and relocated to the peninsula to serve in the ongoing war with the neighbouring Muslim rulers. This shouldn't be seen as strictly a religious conflict, though, as many of these rulers were more interested in the possibility of carving out a bit of land for themselves amidst the chaos of war than they were in overthrowing a rival religion.

In fact, there was a fair amount of intermarriage between different groups on the peninsula. This was most pronounced during the Taifa period, when everyone was making alliances with everyone else to backstab anyone they could, but they existed throughout the Middle Ages. And, of course, in addition to Muslims and Christians there was a substantial Jewish population who also intermixed with both groups. At its best, medieval Iberia was a fascinating mixing pot of peoples and beliefs, while at its worst it saw cases of egregious persecution and violence against the group(s) with the bad luck to be out of power.

Now, I'm not a geneticist, so take this all with a huge grain of salt, but given the vast diversity of people's that have passed through the Iberian peninsula I wouldn't be surprised if some people who lived there were of Arabic descent at the time Spain was invading the Americas, but I would be very surprised if there were so many of them that they had a significant impact on the genetic makeup of American populations. The Arabs were only ever a small minority on the Iberian peninsula - even when they ruled the majority of it - and I'm doubtful that they could have had that much of an impact on the population, especially centuries after their dynasties had all disappeared.

If you want to read more about Muslim, Christian, and Jewish relations in this period, Brian Catlos' Infidel Kings and Unholy Warriors is a very accessible and fascinating book and well worth a read.