My interest is basically historical and I did a little research as to the reason of Friday as public holiday in most of the Muslim countries.
Concept of the day of rest or Sabbath does not exist in Islam and there is no compulsory weekly day-off or holiday as such. Traditionally only Eid ul Fitr & Eid-al Adha were the days when Muslims did not normally work or trade.
Hijra calendar was by the 2nd Khalifa Hazrat Omer (RA) in 639 AD by working backwards to when t Hijra actually took place (622) as year 1. It is not certain when the Islamic New Year became an off –day, in many Islamic countries it is still not.
10th of Moharram as a day of mourning may have been observed earlier privately by the Shiites; however, historical records suggest that during the Caliphate of Al Muqtadir ( 946-974), political power was in the hands of the Buyids dynasty who instituted the formal Ashura observance in 963 AD
According to Al Tabari, there was no formal Friday prayer during the Mecca period. Public Friday prayer was instituted after the Hijra. Understand that Friday was chosen because it was the weekly market day when Jews of Medina bought their provisions for the Sabbath. It has also been suggested that the timing of the Friday (Noon) is roughly the time when the market would normally break and people went home to eat and for the afternoon Siesta.
In Surat at Taubah there is mention of Friday prayer:
(9) O true believers, when you are called to prayer on the day of the assembly, hasten to the commemoration of God and cease trading. This is better for you, if you have understanding.
(10) Only when prayer is ended, scatter in the country and ask for the bounty of God: commemorate God frequently, so that you may prosper.
(11) However, when they see any business or amusement, they flock there and leave you standing. Say: that which is with God is better than amusement and business; and God is the best supplier.
Since the above verse says stop trading, it implies that people had been trading / working before the call to the prayer.
In my personal opinion, practice of the weekly day-off is copying of the Sabbath tradition of weekly holiday and started much later, probably during the Abbasside times. Most Muslim countries including Saudi Arabia now observe Friday & Saturday as weekly off-days.
In conclusion: Friday weekly holiday is not an essential part (Rukn) of Islam. As long as there is time-off for the Friday prayers, the country is free to have any day as weekly holiday without breaking the Islamic law or Tradition of the holy Prophet (PBUH). Hence there is nothing wrong with having Saturday & Sunday as weekly day- off in Pakistan.
All other Islamic holidays are later additions and hence optional.