The formulas for calculating the frequency that a simple string of a given length, wire density, and tension are well known, and are sufficiently accurate to do what is asked. The difficulties arise from the application of the formaulas to any real physical system.

If you suspend a weight on the end of a wire, so that the tension is constant, the wire is stretched so that it's length is increased. When the wire stretches, it's diameter is reduced since there's still the same amount of "stuff" in the wire and the wire is now a little longer. If the diameter is reduced, then the critical parameter of "mass per unit length" is changed since the mass/weight of the string is now "stretched" over a longer length. This relationship is slightly non-linear for real materials, but is accurately described by the "Poisson contraction" that is well documented in elementary level mechanics (strength of materials) handbooks. For accuracy comparable to what is "detectable" with common tuning methods, the specific "Poisson ratio" for each material used in the string will be required, rather than the "about x for most materials" assumptions commonly used for strenght calculations.

If the string is simply hung vertically with a dead weight on the end, you don't really have a "string," you have a pendulum. Different (but still accurate) formulas apply. If you "fix" the length of the "active part of the string" using a bridge and nut as is common on instruments, the friction in the bearing between string and nut introduces an unpredictable difference between the "dead weight" tension and the "actual tension' in the active part of the string. Complex mechanisms have been designed to accomplish "frictionless" nuts, but such mechanisms add their own complications. If you use the common "roller" method, then the "edge" on which the string bears isn't a "sharp" one, and the "effective length" of the thread varies slightly with the amplitude of string vibration. (This probably affects the linearity of the string's harmonics more than it affects the fundamental pitch.)

With typical tuning machines, you tune the string by pulling part of the wire out of the "active length" between the bridge and nut. When you pull on the wire, you increase the tension; but you also reduce the "amount of string" - i.e. the mass of string between nut and bridge. The increase in pitch is caused partly by the increase in tension and partly by the reduction in the weight of the active part of the string. In many configurations the change in tension "dominates the equation" but in some configurations the effect of the reduction in string weight can become "large" compared to the effect of tension change. Typical acoustic string instruments, with modern metal strings, are usually tuned near the "80% of yield" point where the "mass effect" becomes comparable to the "tension effect." The "overtuned" point where twisting a tuning knob doesn't change the pitch happens when the effect of reducing string mass/weight equals the effect of increasing string tension and usually appears just about the time the string breaks.

While the basic equation relating pitch to string length, weight, and tension remains accurate, for a "real world" tuning, additional equations are needed to relate the change in weight and change in tension simultaneously to the change in length - which is what you really change with tuning machines. The needed relationships will be at least slightly different for each string, and possibly for each sample of a given kind of string. These "supplemental effects" will also be affected by temperature, and will change with "ageing" of the string(s). Aging effects can consist of work hardening from repeated stretching, changes in material due to surface and/or internal chemical reactions, changes in "material" due to addition of surface contaminants, etc.

Once the "analysis" for the string is done, the same analysis, with the same accuracy, must be done for the structure that supports the string and maintains the length/tension of the string.

The short answer to the original question is "Of course you can."
If one asks "Is it easy" the answer is NO.
If one asks "Is it useful" then one must revert to the old military "It depends on the tactical situation and the weather."

If one assumes a "musical instrument context," then the answer is that it's a lot easier to "twist the knob until it sounds right."

John