Writing on Facebook about the August 17 incident, Sakina said the police told them: "A passenger on your flight has claimed that you three are members of ISIS... They saw you with Arabic or praise be to Allah on your phone."
Sakina, a clinical pharmacist at University College London, wrote: "Firstly, that's part of the Qur'an, our religious text, so even if we did have it, it wouldn't signify that we're a part of ISIS at all, but regardless, we've had nothing on our phone remotely Arabic related this morning. Also, we're Indian by ethnicity, so we wouldn't even have Arabic in conversation with anyone."
She added: "What are my rights? We would only have been allowed back on the plane if there wasn't a shred of doubt on their part, so someone must be the liar here, in which case, why were those passengers not removed for wasting police time, LYING, making false allegations and racial profiling?"
The siblings were told they could be the subject of further background checks before being allowed to continue their journey.
easyJet has apologised to the siblings for the inconvenience.
A spokesman said: "A member of ground staff requested the assistance of the police who took the decision to talk to three passengers at the bottom of the aircraft steps, before departure.
"The police then confirmed to the Captain that the passengers were cleared to complete their journey and they re-boarded the aircraft and the flight departed to Naples.
"The safety and security of its passengers and crew is our highest priority which means that if a security concern is raised we will always investigate it as a precautionary measure.
"We would like to apologise for any inconvenience caused to the passengers."
A spokesman for Essex Police said: "Essex Police were contacted just after 5.40am on Wednesday, August 17, with reports of concern regarding the behaviour of three people who were looking at their mobile phones.
"Officers at the airport spoke to them and examined their phones with their consent. They were quickly able to establish that no offences had been committed and they boarded their flight. We are satisfied the call was of good intent."