The two probationary officers, aged 23 and 24, appeared before Assistant Chief Constable Robin Smith, who recommended they should be dismissed.
Both officers, who were not identified, have since resigned, Sussex Police said.
The investigation came after they filmed themselves near the site of the air crash where 11 people died and 16 others were injured when a vintage Hawker Hunter T7 jet crashed on to a highway during the Shoreham Airshow on August 22.
Sussex Police had launched an internal investigation after the footage and message were sent on social media to a colleague who reported the pair, the BBC reported.
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The force said both constables exercised their right to tender resignations under regulation 13, a provision specific to probationary officers.
Chief Constable Giles York said the pair were "clearly remorseful" and had acknowledged the impact of their actions, but would not go on to become "well-conducted officers" in his view.
"When it really mattered the officers failed to show understanding and compassion for the victims and their families," York said.