Tsunamis are geophysical rather than meteorological phenomena. These are two separate but related branches of the earth sciences. Their definitions paraphrased from wikipedia are below:
Meteorology is the interdisciplinary scientific study of the atmosphere that focuses on weather processes and forecasting bound by variables including temperature, pressure, water vapor, and the gradients and interactions of each variable, and how they change in time.
Geophysics is the study of the Earth or any other planet by quantitative physical methods, especially by seismic, electromagnetic, and radioactivity methods. It calls heavily upon other fields of science such as math, physics, and computer science.
Atmospheric science (and by inclusion meteorology) is a subdiscipline of geophysics. Thus, all weather is geophysical but not all geophysical phenomena are weather. Tsunamis are mainly created by tectonic movements due to earthquakes, landslides, and volcanoes. Thus, the geophysical subdicipline of seismology is closely linked to tsunami science. Tsunami wave propagation in the ocean is described by physical oceanography, which is another geophysical subdicipline.
Meteotsunamis are the exception to this point of view. These are tsunami-like waves generated by large atmospheric pressure disturbances such as atmospheric gravity waves. They generally have shorter periods of oscillation than tsunamis.
The moral of this story is that, with the exception of meteotsunamis, tsunamis are not related to the weather. Sometimes I think it's a little misleading to the public that the tsunami warning centers happen to fall under the National Weather Service organization umbrella.
Disclaimer: The views expressed in this post are the author's opinions only and don't necessarily reflect those of NOAA or the NWS.