The Hindu equivalent for the modern weekday is called as vāsara or vāra in the modern usage.
Ravi-Vāsara, Som-Vāsara, Mangal-Vāsara etc, which are equivalent of Sunday, Monday and Tuesday do bear a striking resemblance etymologically.
Although Vāsaras do not serve any computational purpose, unlike Tithis, which are used to determine a day in the lunar calendar, they still do form one of the five angas of the Hindu Panchanga, the other four being tithi, nakshatra, yoga and karna. They are also mentioned in the Vedanga Jyotisha which is considered to be the origin of Hindu Astrology.
As for what could be the origin of this system, the answer is the Navagrahas: Surya(Ravi, Sun),Som(Moon),Mangal(Mars), Budha(Mercura), Guru(Jupiter), Shukra(Venus), Shani(Saturn), Rahu and Ketu.
The first seven preside over individual Vāsaras while Rahu and Ketu are responsible for the Solar and Lunar eclipses. All these nine Grahas are said to move with respect to the fixed constellations in the zodiac, hence influencing the lives of the people affected by the respective constellations. It is not very unlike the western belief in astrology where different deities were said to preside over the opening hours of the days named after them.
So it is not just a westernised version of our own system. Tithis and Vāsaras exist independently of each other and fulfill different requirements. Tithis have more significance in the field of Astronomy for maintaining calendars and predicting celestial events like eclipses, Vāsaras have Astrological significance, helping determine the effect of the current arrangement of stars on the life of an individual.
But as with everything else in Hinduism, Astronomy and Astrology are closely related and these two are easily mixed up with one another. It is better to keep track of tithis for the purposes of a calendar of events.