To strictly fit the format, you could use thirst in its verb form. Hence
He thirsted to death.
However, its use as a verb is relatively rare (indeed, it may well be one of those interesting cases where metaphorical use out-numbers literal, and The athlete thirsted for Olympic gold seems more natural than I thirsted for a decent craft-brewed beer, though both events described are as likely to occur).
For that reason it's technically correct, but sounds unnatural, and I wouldn't recommend it. I'd go for keeping He died of thirst.
Hypernatraemia caused by dehydration as given in other answers, are the most likely direct causes of what actually dealt the body its final irrecoverable blow, but there's no verb form of hypernatraemia, and He dehydrated to death has the same problems as thirst as a verb, and is imprecise: diarrhoea is the second biggest cause of infant deaths, and a common cause of older deaths, and then it is also death by dehydration, but not of thirst. Indeed, a patient of dehydration may find it hard to drink as much as their carers are encouraging them to, as while they are dying from dehydration, they are not thirsty.