Two people have been detained so far. One of the suspects said more than 30 men had sex with the girl, but denied rape, according to local media reports.
The men are accused of raping the teenager while she was intoxicated.
Israeli Prime Minister Benjamin Netanyahu called for those responsible to be "brought to justice".
"This is shocking, there is no other word. This is not only a crime against the girl, this is a crime against humanity itself that is worthy of all condemnation," he tweeted on Thursday.
The teenager filed a complaint to police last Friday.
According to Israeli media reports, she went to Eilat earlier this month with a friend, and met up with a group of the friend's acquaintances.
They all went out to drink together before going back to a hotel, where the men allegedly raped the girl, one after the other.
Authorities say the teenager's friend tried to help her, but was not able to stop the men.
The identities of the two detained suspects have not been made public, but local media reports say they are Israeli residents in their 20s.
One was arrested after police found video documentation of the alleged assault in messages between him and the teenager, the Times of Israel reports. The suspect has denied communicating with the girl, and says someone else was using his phone.
The same suspect told police that more than 30 people were involved in the incident, but that security camera footage would prove that it was consensual.
Defence Minister Benny Gantz has condemned the alleged attack.
"The police investigation is still under way, but it's important that I convey a message to those who participated or bore witness to this case or others: The only thing you've proven is what depravity of your soul and morality you are capable of," he tweeted.
"To the complainant, it's important that I say: My heart is with you, you are not alone."
Last year, a 19-year-old British woman claimed she had been gang-raped by 12 Israeli men at a hotel in Cyprus. She was later found guilty of lying about being raped and given a four-month suspended sentence.
The case was denounced by women's rights groups. The 19-year-old's lawyer said she was appealing against her conviction.
SOURCE: BBC
