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On 2020 February 24, during their third observing run ("O3"), the Laser Interferometer Gravitational-wave Observatory and Virgo Collaboration (LVC) detected S200224ca: a candidate gravitational wave (GW) event produced by a binary black hole (BBH) merger. This event was one of the best-localized compact binary coalescences detected in O3 (with 50%/90% error regions of 13/72 deg$^2$), and so the Neil Gehrels Swift Observatory performed rapid near-UV/X-ray follow-up observations. Swift-XRT and UVOT covered approximately 79.2% and 62.4% (respectively) of the GW error region, making S200224ca the BBH event most thoroughly followed-up in near-UV (u-band) and X-ray to date. No likely EM counterparts to the GW event were found by the Swift BAT, XRT, or UVOT, nor by other observatories. Here we report on the results of our searches for an EM counterpart, both in the BAT data near the time of the merger, and in follow-up UVOT/XRT observations. We also discuss the upper limits we can place on EM radiation from S200224ca, and the implications these limits have on the physics of BBH mergers. Namely, we place a shallow upper limit on the dimensionless BH charge, $\hat{q} < 1.4 \times10^{-4}$, and an upper limit on the isotropic-equivalent energy of a blast wave $E < 4.1\times10^{51}$ erg (assuming typical GRB parameters).