We present a sensitive search with the Karl G. Jansky Very Large Array for the radio counterpart of the gravitational wave candidate S191216ap, which is classified as a binary black hole merger and suggested to be a possible multimessenger event, based on the detection of a high-energy neutrino and a TeV photon. We carried out a blind search at C band (4–8 GHz) over 0.3 deg2 of the gamma-ray counterpart of S191216ap reported by the High-Altitude Water Cerenkov Observatory (HAWC). Our search, spanning three epochs over 130 days of postmerger and having a mean source-detection threshold of 75 μJy beam−1 (4σ), yielded five variable sources associated with active galactic nucleus activity and no definitive counterpart of S191216ap. We find <2% (3.0% ± 1.3%) of the persistent radio sources at 6 GHz to be variable on a timescale of <1 week (week–months), consistent with previous radio variability studies. Our 4σ radio luminosity upper limit of ∼1.2 × 1028 erg s−1 Hz−1 on the afterglow of S191216ap, within the HAWC error region, is 5–10 times deeper than previous binary black hole (BBH) radio afterglow searches. Comparing this upper limit with theoretical expectations given by Perna et al. for putative jets launched by BBH mergers, for on-axis jets with energy;1049 erg, we can rule out jet opening angles ≲ 20° (assuming that the counterpart lies within the 1σ HAWC region that we observed).