According to reports, 12 of those arrested were Pakistani, with some of the nationalities of other attackers remaining unclear.
In a statement, the Saudi Interior Ministry said 26-year-old Nayer Musallam Hammad Al-Nijeidi Al-Bluwi, carried out the attack near the Prophet's Mosque in Madinah, which left four security force members dead.
The ministry's statement added that examination of the remains from the two terror attacks in Madinah and Qatif "revealed the presence of the explosive Nitroglycerin", which it said was similar to the substance found in the scene of the bombing at Soliman Fakeeh hospital parking lot in Jeddah.
Hamad had a history of drug abuse, the ministry said.
The ministry said the Qatif attack was carried out by three "terrorists" including a man named as Abderrahman Saleh Mohammed Al Amr, 23, who it said was known to the security services for taking part in protests.
On Tuesday, officials identified the suicide bomber who struck outside the USA consulate in Jiddah in a third attack on the same day.
A bombing near the USA consulate in Jeddah on the same day killed only the attacker, and no casualties other than the bomber have been reported in Qatif.
The Saudi Arabian authorities have arrested as many as nineteen people including 12 Pakistanis for their alleged involvement in Monday's three bomb blasts which rocked Jeddah, Madina and Qatif.
Two police officers were wounded the Jeddah attack.
Saudi policemen stand guard at the site of a bombing near the United States consulate in the Red Sea port city of Jeddah, Saudi Arabia, on July 4, 2016.
It is also not claimed responsibility for this attack from any group.
King Salman, in a speech on July 5, marking Eid al-Fitr, the holiday that celebrates the end of the Islamic holy month of Ramzan, said a major challenge facing Saudi Arabia was preserving hope for youth who faced the risk of radicalisation.
The armed group, which controls areas of eastern Syria and northern Iraq, has frequently denounced the Saudi monarchy and has claimed previous attacks on Shia mosques in Qatif and elsewhere in the kingdom.