The 23-year-old fishmonger is coupled with 19-year-old Gemma Owen - daughter of footballer Michael Owen - on the ITV dating show, and has attracted criticism for his treatment of the female contestants.
Bish's family wrote on his Instagram account: "I want to apologise on behalf of Luca for his insensitive reaction on last night’s episode. I know that when he watches it back, he will be embarrassed and deeply apologetic.
"We as his family don’t condone his behaviour, but we do understand he is in an intense environment where his emotions are heightened. Last night was not reflective of the boy we all know and love at home."
They added: “Please be aware death threats and nasty comments totally overstep the mark and [are] something we as a family should not have to endure.”
The family went on to say they have turned off comments and messages on social media as a result, but added their thanks to all those who had sent messages of support.
On the latest episode of Love Island Bish was shown encouraging other male contestants to be unfaithful to their couples while the women had been sent to Casa Amor.
In contrast, in a previous episode he reacted angrily and aggressively to discovering his coupling Owen had talked to another man at Casa Amor.
The reality show has attracted accusations of misogyny for shaming the females for sexual encounters, while glorifying the men for the same behaviour.
Leading domestic abuse charity Refuge said the alleged "misogyny and casual sexism" was "extremely concerning".
A spokesperson for Refuge said in a Tweet: "The misogyny and casual sexism witnessed on this series of Love Island is extremely concerning.
"The double standards, gaslighting and coercive control being displayed by the men in the villa is hugely problematic."
This follows the comments of Teresa Parker, head of communications and media relations at Women's Aid, who told Yahoo News UK: "Love Island is watched by many young people and we know what a huge influence it has.
"Producers must make sure there is support for contestants throughout, and intervene if relationships become unhealthy or abusive."
Her comments came after Adam Collard - a contestant from the 2018 series - returned to the villa as a 'bombshell'.
During his original appearance on the show in series four, Collard was criticised by Women's Aid for his alleged "gaslighting and emotional abuse" towards the female contestants.
Watch: Love Island's Luca was furious to learn Gemma had talked to another man