No, standard healing does not do this.
A vampire can't exactly capture a disease in the same way that a mortal can. Even if a vampire drinks infected blood, they themselves do not get sick. Instead, they potentially become carriers for the disease, and could transmit it when it feeds due to things like bacteria or viruses that are in its fangs that are then stabbed in others. The most direct analogy that occurs to me is that it is like passing dirty needles: once contaminated, they can transmit the disease to anyone who uses them.
The standard healing mechanics for non-aggravated damage (spending a blood point, restoring a punch box) does not work for this. Diseases do not really harm the vampire at all, so there is nothing to cure. Vampires generally do not have the ability to disinfect or purify things, so they cannot simply clean the blood. It is not clear how long a disease can persist in this way (an undead body is not 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 is inconvenient and requires access to some unusual Disciplines.
First, you will need someone with Obeah and Valeren at two points each. That person can take the combination Discipline Increasing disease (Lore of the Bloodlines, page 81). Using this is a Perception + Medicine roll on difficulty 7 to learn detailed information about your injuries or illnesses.
Second, that person can use the three-point Obeah power Corpore Sano. Normally this would not cure a disease, even if it is identified using Sense of vitality, why Corpore Sano It can only restore health levels (as does the use of blood to heal). But Increasing disease state
augured, the vampire can use Corpore Sano (V20, p. 469)
cure the disease directly, on a difficulty determined by
The community and the 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).
There you go! To my knowledge, there is no reason for anyone to use these powers in themselves. So, if you are a vampire with the correct Disciplines, or know someone who knows, it is possible. The right disciplines are difficult. Obeah is in the clan only for Salubri, while Valeren is in the clan only for Salubri antitribe.
Before I could review my books, I mistakenly suggested the Tremere thaumaturgy ritual Purity of meat. This ritual explicitly does no Cure blood diseases.
The ritual I was thinking about is of Blood magic: secrets of thaumaturgy, page 94, and it is the level 3 thaumaturgy ritual Meat cleaning. However, this is a revised edition source, so it does not apply to V20.
That ritual cures the vampire of any drug addiction or non-supernatural diseases. It also requires the subject to lose all but one point of blood at a rate of 2 points per hour, so it remains largely an option only for vampires.