Yes, in principle.
What you're describing is called "relative pitch": the ability to determine the interval between two notes. Developing relative pitch is a standard part of much music training at the collegiate level, and many people develop the skill earlier.
Typically this is trained by learning the distinct sound of each type of interval, allowing its identification without knowing the specific identity of either pitch. However, the ability to recognize intervals would also mean that, given a known fixed pitch, another pitch could be identified by ear.
Most training courses limit themselves to intervals up to and including an octave. Some go beyond that by a few notes. Intervals larger than an octave — "compound intervals" — are not usually taught, because the sound sufficiently similar to their "simple" counterparts, function musically in the same way, and the goal of relative pitch is interval identification rather than pitch identification.
However, one can learn, at least in principle, to identify compound intervals with specificity, and this is what would allow for the kind of pitch identification being suggested in the question.
There are numerous questions on this site regarding the development of this ability. Search the tag relative-pitch