You have not mentioned the rank of your father. An officer's case of this type is treated more seriously than that of a person below officer rank.
There are several opportunities given in such a case before a court marshal is ordered. In the stage before a court martial, which is a summary of evidence the accused is given an opportunity to present any evidence that may prove that he had to stay back because of a medical emergency pertaining to a close family member and that he did send a request for extension of leave, if any. If no extension was possible as full entitlement of leave had already been availed, the Army authorities have to proceed with the case and after a minor administrative action or condonign the deemed offence, they would regularise the period of absence without leave as Extra Ordinay Leave without pay.
There are several such cases each year below officer rank and the action taken is such cases is quite minor, As such your father should not think that his pride will be affected or that anyone will doubt his sincerity.