Welome to Sclero Forums. I'm sorry that you have multiple sclerosis, and onycholysis (the formal name for nails separating from the nail bed.)
As it happens, it's very odd that your doctor would suspect scleroderma, based only on that. It would be far more commonly caused by trauma, tapping on keyboards, manicures, soaking hands in water too long, fungal infections, psoriasis, certain medications (especially some antibiotics), anemia, and hyperthyroidism. And, as Jo pointed out, onycholysis is not part of the diagnostic criteria for any type of scleroderma (either localized or systemic).
Unless your doctor has additional reasons to suspect scleroderma (or lupus), I'm sorry they have put you through this worry, which is unnecessary in the absence of other specific symptoms such as Raynaud's, tight skin, and pulmonary fibrosis. Of course, your doctor might be on guard because of your family history, and because very rarely, multiple sclerosis can occur in overlap. If this is just a single symptom, and otherwise you don't have things outside of MS, then I'd look in other directions for a cause of this.
I have scleroderma in overlap, and I've had onycholysis, too (two fingers, one on each hand, at the same time) -- but it was chalked up to psoriasis (which I also have), and I seriously doubt that the thought of scleroderma as a differential was even briefly entertained. In fact, it never even crossed my mind, and it wouldn't have unless I was also having significant sclerodactyly and prominent Raynaud's attacks, etc. at the same time.
Even in a person already diagnosed with scleroderma, it should still be the last culprit considered for any new symptom, because all the normal or more common causes need to be ruled out first. Often, more common causes are also more treatable. Just think, this could also be a symptom of hyperthyroidism or anemia, both of which can also be associated with MS.
If it's a one-time problem, I wouldn't worry about it, so much as if it recurs. But I'm not a doctor, I have no medical training at all, so you should absolutely bear in mind that your doctor is much better equipped to assess your health, and suspect and rule out various ailments, as we are here on the forums!
I just want you to be assured that, if this is the only reason they suspect scleroderma (and I actually rather doubt it is in your case; I am aiming this at others who stumble upon this thread who only have onycholysis), it would be insufficient, by any standard that I'm aware of, which should hopefully set your mind at ease if or until any other symptoms outside of MS develop.