Activists have urged the Government to support Ms Alqunun in her bid for asylum in Australia, and said the young woman should be issued with emergency travel documents.
However, officials in Australia hinted that her request will be accepted.
The UNHCR has since assessed her case and found she is a refugee.
Saudi Arabia described the case as a "family affair", adding that the country did not demand her deportation back home.
Rahaf Mohammed Al-Qunun in Bangkok. "And I would have preferred it better if her phone was taken instead of her passport".
Rahaf Mohammed al-Qunun, 18, arrived in Bangkok from Kuwait late Saturday after slipping away from the family she has accused of abusing her. Qunun said it was her intention from there to seek asylum in Australia, but she was stopped by officials in Thailand, who confiscated her passport. But who is Rahaf al-Qunun and why is her life in danger?
In an earlier and separate explanation released on Twitter, the embassy also denied sending officials to Suvarnabhumi airport to meet Qunun as she arrived via Kuwait or impounding her passport - as she alleged. "The Saudi charge d'affaires said he is satisfied and expressed confidence on the work of Thai immigration, of the Thai government, and of the Foreign Ministry yesterday".
"She is now under the sovereignty of Thailand; no-one and no embassy can force her to go anywhere", he said. She said access to twitter had "changed the game" in what was wished for her.
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Ms Alqunun's case has again highlighted the cause of women's rights in Saudi Arabia.
According to the worldwide law's principle of non-refoulement, asylum seekers can not be returned to their country of origin if their life is under threat.
Thai officials met with counterparts from the Kingdom on Tuesday and Maj Gen Surachate Hakparn, Thailand's immigration police chief, said that Saudi officials said they were satisfied with the handling of the case.
Her father, a Saudi government official, and brother landed in the capital last night and immediately asked to see Ms Alqunun, AAP reports. Those women and the men who are fighting for women's rights activists who are in prison today in Saudi Arabia, they are the leaders - the vanguard of a revolution that will free Saudi women... by ending once and for all the guardianship system which is the foundation of patriarchy in Saudi Arabia...
She made her escape during a family trip to Kuwait, where she purchased flights to Thailand and Australia. Thai immigration authorities have said that al-Qunun was prevented from onward travel to Australia, having been in contact with the "Saudi Arabia embassy to coordinate" this.
When someone makes very serious allegations of abuse, torture or a threat to their life, authorities have to take it at face value and allow the proper protocols of investigation to occur, Ms Stirling said. Alqunun's family was abusive, she said, even more so since she had renounced Islam.
Some Saudi female runaways fleeing abuse by their families have been caught trying to seek asylum overseas in recent years.
For more than 24 hours she fed global media, rights campaigners and a concerned public a stream of dramatic videos and descriptions of her ordeal.
The ultra-conservative Saudi Arabia has come under fire since the murder of journalist Jamal Khashoggi in the country's consulate in Istanbul a year ago.