No, your facial expression has no direct correlation with the fundamental frequency you vocalise. However, smiling while singing does in fact tend to have a number of favorable side effects.
Traditionally, when I've heard singing (usually choral) instructors explain this, they say that smiling will prevent you from going flat on a note, and that it also prevents you from being monotonous/lifeless. And from my own personal experience, it's seemed to largely be effective advice, especially at lower levels of singing, where the singers may not be as skilled at sustaining a note and keeping it in tune every single time.
If I had to guess (and that's all this is, a somewhat educated guess), I'd say that somehow the act of smiling is closely linked to certain elements of the singing mechanism that are beneficial, and therefore thinking about smiling sort of unconsciously enables the singer to produce the sound more optimally (hence this "sharpening" would really just be a better sound production mechanism being better at singing in tune). And obviously, thinking about smiling can help one be more emotionally aware of the music.
So yes, other answers are correct in that a singer can frown and be perfectly in tune, or smile and be off by half an octave. But choral directors and other singing instructors have been conscious of the correlation between the two human processes, and as such, it makes sense to use this relationship in an educational setting in order to facilitate better singing technique.
Of course, I'm not a vocal instructor myself, so if you're looking to use this idea in your own singing studies or teachings, please talk to actual professionals about how to learn or teach the specifics of this - like many singing ideas, if you do it wrong, it may be more harm than good.
A couple random online sources I found discussing this: