Determine sleep apnea from a sound recording?

Not as far as I know, but it isn't inconceivable. As far as I am aware, only a blood O2 sat monitor, say a finger sensor, maybe a 'snore app' listening to you all night (snoring is associated with sleep apnea), and some kind of heart rate and rhythm monitor are about all that might paint a coherent picture. Otherwise, it's a sleep lab or some other device and leads/sensors. The most reliable way is to actually measure the rate and volume in breathing, and that means a mask, fitted reasonably well, to measure tidal volume, flow limitations, etc.
If you know for a fact, or if a partner reports to you, that you gurgle, snark, gasp, and otherwise seem to starve for air routinely each hour when you sleep, then yes, but the experts prescribing care need to know veridical numbers.

I don’t know about them providing any diagnosis, but the apps can record and rate your snoring, Snore Lab is one such app. It picks up on everything. So, if you fluff your pillow, it’ll show as noise.

@robertwills
It may help you know whether you snore or gasp for breath while sleeping if you sleep alone.