Answer: There's a web site for everything (and this is about to become one for this subject). If you build a triad on each degree of the natural scale, using only the pitches available in that scale, you'll end up with the following (let's assume I started with C as the tonic, as you request):

C minor, D diminished, Eb major, F minor, G minor, Ab major, Bb major

In functional harmony those are i, ii dim, III, iv, v, VI, and VII:

Normally in harmonic music the V will be converted to a major triad, borrowing a note that is outside the natural scale but found in the major scale: in this case, B natural to make G major. Then you'd have i, ii dim, III, iv, V, VI and VII.

In theory you could follow a similar principle with any scale. But some scales are not really designed for harmonic music. Traditional Arabic music, for example, is basically melodic; it doesn't deal in functional harmony that can be represented with I, ii, iii, etc. Music traditions that emphasize melody often instead make use of drones or parallel intervals.

If you want to depart from those traditions and mingle eastern scales with western harmony you could treat, for example, the Persian minor scale analogously to the minor scale shown above. Using the typical interpretation of the Persian minor but starting on C again you'd get C, Db, E, F, Gb, Ab, B. But if you try building triads on each of those tones while using only the pitches available in the scale, you quickly see how this scale doesn't easily fit into the tradition of triadic harmony. On C you'd get C diminished, on Db you'd have Db major, but on E you'd have a triad consisting of E, Gb, and B. That is neither major nor minor; it doesn't fit into the triadic system. Similarly, the chord on Gb would be Gb, B, Db, which is neither fish nor fowl:

If you alter those chords to make them more normal then you're departing from the Persian minor. A similar process is what led the medieval modes Dorian, Phrygian, Lydian, Mixolydian, etc. to coalesce finally into major and minor.