Context. To constrain present star formation models, we need to simultaneously establish the dynamical and physical properties of disks and jets around young stars. Aims. We previously observed the star-forming region G16.59−0.05 through interferometric observations of both thermal and maser lines, and identified a high-mass young stellar object (YSO) which is surrounded by an accretion disk and drives a nonthermal radio jet. Our goals are to establish the physical conditions of the environment hosting the high-mass YSO and to study the kinematics of the surrounding gas in detail. Methods. We performed high-angular-resolution (beam FWHM ≈ 0′′.15) 1.2-mm continuum and line observations towards G16.59−0.05 with the Atacama Large Millimeter Array (ALMA). Results. The main dust clump, with size ≈10⁴ au, is resolved into four distinct, relatively compact (diameter ~2000 au) millimeter (mm) sources. The source harboring the high-mass YSO is the most prominent in molecular emission. By fitting the emission profiles of several unblended and optically thin transitions of CH₃OCH₃ and CH₃OH, we derived gas temperatures inside the mm sources in the range 42–131 K, and calculated masses of 1–5 M_⊙. A well-defined Local Standard of Rest (LSR) velocity (V_LSR) gradient is detected in most of the high-density molecular tracers at the position of the high-mass YSO, pinpointed by compact 22-GHz free-free emission. This gradient is oriented along a direction forming a large (≈70°) angle with the radio jet, traced by elongated 13-GHz continuum emission. The butterfly-like shapes of the P–V plots and the linear pattern of the emission peaks of the molecular lines at high velocity confirm that this V_LSR gradient is due to rotation of the gas in the disk surrounding the high-mass YSO. The disk radius is ≈500 au, and the V_LSR distribution along the major axis of the disk is well reproduced by a Keplerian profile around a central mass of 10 ± 2 M_⊙. The position of the YSO is offset by ≳0′′.1 from the axis of the radio jet and the dust emission peak. To explain this displacement we argue that the high-mass YSO could have moved from the center of the parental mm source owing to dynamical interaction with one or more companions.