There is lactose in milk-based formula as well, so it's not likely the case. I do know that the switch from formula to cow's milk can cause looser stools - a totally normal reaction that some LOs may have, others may not, but it's not considered abnormal...just the body getting used to the change. (Info is from our hospital paed when we saw him about DS doing a milk challenge to see if he had outgrown his milk allergy - he told me that when moving on to cow's milk, you can't judge diarrhea ALONE as an indicator that there is still a milk allergy, because the switch very often causes diarrhea while LO gets used to it).
If he is not in any discomfort and he is happy to drink the cow's milk, I would continue on with it and just keep your eye on his poop. If it persists for a week or so, it's worth a phone call to your LO's doctor just to see what he/she advises.