No, standard healing does not do this.
A vampire can't exactly catch a disease in the same way that a mortal can. Even if a vampire drinks infected blood they themselves don't become ill. Instead they potentially become carriers for the disease, and could pass it on when feeding due to things like bacteria or viruses being on their fangs which are then stabbed into others. The most direct analogy I can think of is it's like passing around dirty needles-- once contaminated, they can spread disease to anyone that uses them.
The standard healing mechanic for non-aggravated damage (spend a blood point, restore a hit box) doesn't work for this. The diseases don't actually damage the vampire at all, and so there is nothing to heal. Vampires do not generally have the ability to disinfect or purify things and so they can't just clean the blood. It's not clear how long a disease can persist this way (an undead body isn't the ideal environment for infectious bacteria and viruses).
In V20 this is difficult to do, but possible.
I had to dig deep to find an explicit way to cure a disease! It's inconvenient and requires access to some unusual Disciplines.
First, you will need someone with both Obeah and Valeren at two dots each. That person can then take the combination Discipline Auguring the Sickness (Lore of the Bloodlines, page 81). Using this is a Perception + Medicine roll at difficulty 7 to learn detailed information about their injuries or illnesses.
Second, that person can then use the three dot Obeah power Corpore Sano. Normally this would not cure a disease, even if identified by using Sense Vitality, because Corpore Sano can only restore health levels (like using blood to heal does). But Auguring the Sickness states
augured, the vampire may use Corpore Sano (V20, p. 469)
to cure the disease outright, at a difficulty determined by
the commonality and virulence of the disease (difficulty 5
for the common cold, difficulty 7 for most forms of herpes
or chicken pox, or difficulty 9 for Ebola or HIV).
So there you go! As far as I know there's no reason someone couldn't use these powers on themselves. So if he's a vampire with the right Disciplines, or knows one who does, it's possible. The right Disciplines is tough. Obeah is in-clan only for Salubri while Valeren is in clan only for Salubri antitribu.
Before I could check my books I had mistakenly suggested the Tremere Thaumaturgy ritual Purity of Flesh. This ritual explicitly does not cure blood diseases.
The ritual I was thinking of is from Blood Magic: Secrets of Thaumaturgy, page 94, and is the level 3 Thaumaturgy ritual Cleansing of the Flesh. This is, however, a Revised Edition source, so it doesn't apply to V20.
That ritual cures the vampire of any drug addictions or non-supernatural diseases. It also requires the subject to lose all but one blood point at a rate of 2 points per hour, so it's still largely a vampire-only option.