Discharge isn't used that way in this context.
While you can make an argument for using the word this way not being grammatically incorrect, technically, this is not the way that a native English speaker would describe the event of being released from the hospital (not in the U.S., at least).
The usage you suggest is common when discussing terms of military service ending, particularly if describing the conditions of the end of service:
He received an honorable discharge from the Air Force.
Members of the military have an explicit obligation to do certain things and behave in certain ways while they are in the military, and the conditions under which they leave military service can continue to apply to their lives. For example, if someone is dishonorably discharged from the military they are legally required (in the U.S.) to say so when applying for jobs in the future.
So, while a person may have been discharged from the military, it may be the case that they also "have a discharge" which continues to describe them in the present.
Usage of discharge in this way is strongly related to employment, specifically (see definition 8).
The constructions in the question are a somewhat different situation. Charge is, in terms of a hospital stay, about a person being given over to the hospital for care. It is true that that person is a charge of the hospital. But discharge doesn't describe that person or an event which continues to be relevant to their identity into the future. Instead it only describes something that happens to them: they are, themselves, discharged from the hospital. The interpretation of discharge here is that they physically leave the hospital.
Further, a discharge is typically not something that a hospital can grant or withhold-- nearly all hospital stays are voluntary on the part of the patient, who can leave at any time they wish. Circumstances in which the patient may not choose to leave are generally described as being committed to the hospital, and a commitment ending is described as such (or the patient is released from the hospital, emphasizing that they were being held there previously).
In that light it makes less sense to describe someone as "getting" a discharge from the hospital, as it is not something the hospital can give, or decline to give.
I can't promise that there are no English-speaking regions where people use discharge as in the question, but it's definitely not typical usage. Whether or not it is technically acceptable according to a particular standard of grammar, this is not a good construction to convey your intended meaning in English.
If we leave these considerations aside, the constructions would be broadly OK. Try substituting a word used more typically as a concrete noun in such a case:
He got a bottle of medicine from the hospital.
He has got a bottle of medicine from the hospital.
These are correct because the bottle of medicine is a noun, which he received in the past. The second version indicates that he received the bottle of medicine in the past, continued to have it from then until the present, and still has it today.
Since discharge is not commonly used as a noun in this scenario (even if it technically could be so used) native speakers will tend to think of it as an action strictly in the past which may or may not apply to the present:
He has gotten a discharge from the hospital
But again, this is really an idle question. Technically correct or not, the constructions in the question simply aren't used in English.