Does the unit of measure matter when you are solving for the diameter of a circumference?

Both the diameter and the circumference are lengths, so it doesn't matter what units you use to measure them, as long as you are consistent: both in cm, or both in in., or both in miles or... .The ratio of circumference to diameter will then be $\pi$. In your example, a circle of circumference 10in is bigger than a circle of circumference 10cm; the diameter of the bigger circle is $10/\pi$ in., so the ratio is $\pi$, and the diameter of the smaller circle is $10/\pi$ cm, so the ratio is also $\pi$.

In Euclidean geometry the ratio of the circumference to the diameter has the the same numerical value for all circles. We call that number $\pi$. That fact is independent of any way you might measure lengths. Lay the diameter along a string and mark its multiples. Then wrap that string around the circle - it will take $\pi$ diameters to get back to the starting point.

If you simultaneously mark the string in inches (so measuring the diameter in inches) it will take $\pi$ times as many to travel the circle. If you measure the diameter with centimeters (or paper clips, as they do in first grade) you get the same multiple for the circumference.

Note that the ratio of circumference to diameter is the same for all circles only on the plane. For circles on a sphere (where "lines" are great circles) that ratio is smaller the larger the circle. It's always less than $\pi$, but close for small circles.

Related: How to prove that $\pi=$ constant ratio

The diameter of a circle with a circumference of $10$ inches is $\frac{10}{\pi}\approx 3.1831$ inches

Similarly the diameter of a circle with a circumference of $10$ centimetres is $\frac{10}{\pi}\approx 3.1831$ centimetres

You could mix things up, but this is unlikely to help. $10$ inches is $25.4$ centimetres so you could say the diameter of a circle with a circumference of $10$ inches is $\frac{25.4}{\pi}\approx 8.085$ centimetres, but why would you want to?