Army Chief Sondhi Boonyaratglin asked Thailand's military attach in Burma to deliver the letter and notify officials there about the incursion.
The skirmish occurred late Wednesday in northern Mae Hong Son province.
Burma has yet to respond or comment on the Thai soldier's death.
Thai military officials have suggested that the incident may have been a misunderstanding and that Burmese troops may have mistaken the Thai soldiers for ethnic Burmese insurgents.
General Sondhi says that the two sides will hold talks to discuss the incident.