Short answer: no, no such thing exists.
Alternately: does a spectrogram qualify?
Long answer: there are quite a few problems with precise acoustic transcription.
In particular, spoken sound isn't divided into nice distinct units. When an English-speaker says
[khæt], the actual acoustic reality is silence, then a tiny bit of turbulence, then a continuous vowel sound, then silence again. The
[k] and the
[t] don't have any acoustic existence in and of themselves—the only sign of them is how the
[æ] gets warped at the beginning and end.
How would an "acoustic phonetic alphabet" deal with this in a useful way? If you record someone saying
[up], you'll see a different "warping" effect in all four of those, with none of them "warping" in quite the same way. But a human listener will process and interpret all those effects subconsciously without needing to think about it, and tell you that
[uk] have the same consonant, and
[up] have the same consonant, and those two consonants aren't the same.
And articulatory studies show that, indeed, there is something the same between
[uk], and between
[up], and that something is where the closure occurs in the vocal tract—the place where the airflow is cut off. So that's what the IPA represents, not an acoustic fact, but an articulatory one. Not because it's more accurate or anything like that, but because in this case, the articulatory facts are more useful for linguists than the acoustic ones. When precise acoustic details are needed, phoneticians tend to stick to spectrograms, which can represent all those details more completely and precisely than any alphabet could.