Hampshire Police posted a photograph of a red car on its side, asking people to "retweet for a Hyundai i20 or FAV for a Vauxhall Corsa".
Some people called the tweet "inappropriate", but it was also described as "moronic".
"We were simply trying to show the impact of the crash. We didn't mean to offend," the @HantsPolRoads responded.
"Getting some negative replies to crashed car tweet.
"The driver was not injured in this incident. We were trying to highlight how much a car can be damaged in a crash."
No one was seriously injured in the crash on Wednesday on Redbridge Road, Southampton.
A number of people called the post "inappropriate", "irresponsible" and "making light of someone's traumatic experience."
Ian Taylor said: "Not inappropriate in the slightest. Absolutely moronic."
Alex Oates said: "A game out of a crash? What's next? 'did the driver/cyclist/pedestrian live? RT yes, FAV no?'."
Wayne Herridge said he had reported the post to Twitter as an inappropriate tweet.
A spokesman for Hampshire Police said two vehicles were involved in the collision and the driver of the red Hyundai had to be freed by Hampshire Fire and Rescue Service.
Nobody was seriously injured in the crash.

The towns of Virginia Water and Cobham, in Surrey, have become Britain's first million pound towns - where average house prices are more than £1m.
Beaconsfield in Buckinghamshire is also in the millionaire's club, according to research by Lloyds Bank.
They are the first towns outside London where prices have hit seven figures.
The research was based on data from the Land Registry for the first half of 2015.
Prices in Virginia Water - home to the likes of Sir Cliff Richard and Sir Bruce Forsyth - average no less than £1.169m, making it Britain's most expensive town outside the capital.
No wonder that the town's famous golf course, Wentworth, feels able to charge joining fees of £125,000. That is on top of the annual membership fee of £16,000.
Cobham - familiar to Chelsea footballers and their WAGS - has average prices of £1.043m.
And anyone wanting to buy in Beaconsfield can expect to pay £1.003m.
"We're seeing the emergence of towns where the average price is at least £1 million," said Sarah Deaves, private banking director at Lloyds Bank.
"Whilst there are several London neighbourhoods where prices are already at this elevated level, outside of the capital this is a first."
However the figures also show a sharp slow-down in the number of homes sold for more than £1m.
In the first half of 2015 there were 5,599 such sales, down from 6,303 in 2014. That amounts to an 11% fall.
One reason for that is the change in Stamp Duty rates, introduced in December 2014.
The buyer of a £1m house will now pay £43,750 in Stamp Duty, up from £40,000 previously.

The entertainer's funeral will take place on Thursday 20 August at St Mary's Church in the suburb of Woolton.
The Right Reverend Thomas Williams, Auxiliary Bishop of Liverpool and former Parish Priest of St Anthony's, Scotland Road, will officiate.
Black, who emerged from the city's music scene to find fame in the 1960s before becoming a TV host, died after having a stroke at the age of 72.
After the funeral, her body will be laid to rest at a private ceremony in Allerton Cemetery, where her parents are buried.
Her late husband and manager, Bobby Willis, was cremated in 1999.
The late entertainer's representative said further details would be released "once all the preparations have been finalised".
A statement said: "Once again the family have asked for their privacy to be continually respected at this difficult time."
The Blind Date and Surprise Surprise host died of a stroke after falling over at her Spanish home earlier this month.

The 32-year-old has scored one goal in 10 appearances for the League Two side in all competitions this season.
"He has been extremely unlucky with injuries and is going to be a big loss while he recovers," head coach Dermot Drummy told the club website.
"We just hope that when he comes back it he can play a full part for us in the last few months of the season."
People have attacked a tweet by a roads policing unit asking readers to identify the brand of a crashed car.
[NEXT_CONCEPT]
One has a golf course that charges £125,000 to become a member, and the other has a post office said to stock bottles of Bollinger and Dom Perignon.
[NEXT_CONCEPT]
Cilla Black is to be buried in her home city of Liverpool next week.
[NEXT_CONCEPT]
Crawley Town striker Matt Harrold will be out for three months after suffering an Achilles injury.