So the short answer to your question is no, its not a one to one correspondence in the general case.
The long answer
So there is a fundamental differences in how pitches are defined in both western and Indian contemporary systems. In the western system there is a set of "standards" for defining pitches. One such example is the concert tuning : https://en.wikipedia.org/wiki/Concert_pitch.
The most commonly defined standard defines a pitch (lets say A4 = 440Hz) and uses the equal temperament tuning to define the other pitches. The formula is f(n) / f(n - 1) = 2 ^ (1/12). So for instance A#4 = 440 * 2^(1/12) Hz.
This defines an entire set of notes along with its defined frequencies. Some standards differ (some folks like defining A4 as 432 Hz) but the central argument of absolute pitches does not change.
The Indian classical traditions does not have such an absolute frame of reference. A "Sa" can be any real number frequency. Lets say that frequency is f. Then the "Pa" is 1.5 times "Sa" so 1.5f. Each of the notes in the Sargam is some fraction of "Sa" based on just intonation.
So while its possible to define your Sa as C and draw a correspondence (well almost, equal temperament is not a perfect mapping to just intonation), you can also defined your Sa as C4 - 10hz, which does not map to any of the concert pitches. (A notable example would be contemporary sitar players tuning their Sa in between C# and D as their Sa).