We present results of the first four Swift satellite follow-up campaigns seeking to identify transient or variable X-ray or UV/optical sources that might be associated with individual candidate high-energy cosmic muon neutrinos detected by the IceCube Neutrino Observatory. Real-time public alerts providing coordinates and arrival times of likely-cosmic neutrinos have been provided by IceCube, via the Astrophysical Multimessenger Observatory Network, since April 2016. Subsequent Swift X-ray observations of four likely-cosmic neutrinos (events 160731A, 161103A, 170312A, and 170321A) reveal multiple X-ray sources in the targeted 90%-containment regions, most of which have been previously identified, and none of which are considered likely sources of high-energy neutrinos. Observations exclude association with the brightest 30% to 65% of Swift-type gamma-ray burst X-ray afterglows over the observed regions. Contemporaneous Swift UV/optical observations, providing reduced coverage of the event localizations, also reveal no candidate transient or variable UV/optical counterparts. We discuss the results of these campaigns and our plans for further follow-up of likely-cosmic high-energy neutrinos from IceCube.