Why is it unacceptable to say "the range is a function of the domain"? I understand that a function is defined as a correspondence between two sets, the domain and the range.
While the definitions of the domain of  a function and the range of a function are said to be contained within the definition of a function.
Now, when a measurement varies over time we say that it is a function of time. Why is it not generally acceptable to replace the word "measurement" with range and the word "time" with domain, and say that the range is a function of the domain, the same way as we say y is a function of x?

I suggest that you have a look at the wikipedia articles domain, image and codomain/target set. As described in the articles range may refer to image or codomain.
Consider the function $f\colon X\to Y$. $X$ is the domain and $Y$ is the codomain. Not everything in $Y$ might be "hit" by the function. The image of the function is defined as $\{ \, y \in Y \, | \, y = f(x) \text{ for some } x \in A \, \}$.
Example:$f\colon \mathbb{R}\to \mathbb{R}, f(x) = x^2$
The domain and the codomain are $\mathbb{R}$. But the image is $[0,\infty)$.
