The two probationary officers, aged 23 and 24, appeared before Assistant Chief Constable Robin Smith, who recommended they should be dismissed.
Both officers have since resigned, Sussex Police said.
The investigation came after they filmed themselves near the site of the air crash where 11 people died after a jet crashed on to the A27 on 22 August.
The recording was made at the police cordon and accompanied by an inappropriate and offensive message.
Sussex Police launched an internal investigation after the footage and message were sent on social media to a colleague who reported the pair.
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," Mr York said.
Genuine mistakes were forgivable, he added, but said the behaviour of the two officers was "inexplicable and caused unnecessary distress to the families of those involved".
The Sussex Police Federation, which had represented the two officers throughout the case, said: "We are content that they have been judged fairly and accept and understand the decision that has been reached."

The 23-year-old mutually cancelled his deal at the Shrimps earlier this month but can train at the Championship side ahead of the move.
"Agreeing the pre-contract deal now, I get the opportunity to bed myself in before I can play again," he said.
"I think this will be a really good challenge for me."
Morecambe boss Jim Bentley had spoken about selling a player to raise funds for the Lancashire side who paid their players late at the start of November.
Despite cancelling Barkhuizen's contract at the Lancashire club, they retained his rights so Preston had to pay the compensation package.
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The former Oceana nightclub building will be demolished in October and replaced with a new office block.
It forms part of the council's plans to transform the Kingsway into a business district.
The authority is moving from its seafront location as part of a wider regeneration of the city centre. Work is due to start in 2017.
The current civic centre site is likely to be replaced with public spaces, tourist attractions and homes.
Swansea council bought the Oceana building - which was once at the heart of the city's nightlife - in May and it is understood it has been earmarked as the potential new home for the civic centre and central library.
Demolition is due to take until March.
"Once we have control of that site we will look at the development and possibly start construction in 2016 to build offices which could be home for the council offices and Swansea library," said Swansea council leader Rob Stewart.
The building, originally constructed by the Rank Organisation in 1967 for a cinema, has accommodated a number of nightclubs over the years, including Ritzy and Icon, Time and Envy, and most recently, Oceana.
It also previously housed an Odeon cinema and Tesco before they were turned into a different club and bar respectively.
But it is no longer fit for purpose and a council move is one of a number of options being considered for the site.
Two Sussex police constables questioned over their conduct at the site of the Shoreham air crash have left the force.
[NEXT_CONCEPT]
Preston North End have signed forward Tom Barkhuizen from League Two side Morecambe for a compensation fee with the deal to go through on 1 January.
[NEXT_CONCEPT]
The site which used to house Swansea's biggest nightclub could become the new home for Swansea council and library.