Context. The recent astrometric discovery of the nearby (590 pc) massive (33 M-circle dot) dormant black hole candidate Gaia BH3 offers the possibility to angularly resolve the black hole from its companion star by using optical interferometry. Aims. Our aim is to detect emission in the near-infrared K band from the close-in environment of Gaia BH3 caused by accretion. Methods. Gaia BH3 was observed with the GRAVITY instrument using the four 8-meter Unit Telescopes of the VLT Interferometer. We searched for the signature of emission from the black hole in the interferometric data using the CANDID, PMOIRED, and exoGravity tools. Results. With a present separation of 18 mas, the Gaia BH3 system can be well resolved angularly by GRAVITY. We did not detect emission from the black hole at a contrast level of Delta m = 6.8 mag with respect to the companion star, that is, f(BH)/f(star) < 0.2%. This corresponds to an upper limit on the continuum flux density of f(BH) < 1.9 x 10(-16) W m(-2) mu m(-1) in the K band. In addition, we did not detect emission from the black hole in the hydrogen Br gamma line. Conclusions. The non-detection of near-infrared emission from the black hole in Gaia BH3 indicates that its accretion of the giant star wind is presently occurring at most at a very low rate. This is consistent with the limit of f(Edd) < 4.9 x 10(-7) derived previously on the Eddington ratio for an advection-dominated accretion flow. Deeper observations with GRAVITY may be able to detect the black hole as the companion star approaches periastron around 2030.