Just to briefly comment on some things folks have said in this thread:
Tsunamis are expected in the aftermath of large earthquakes that involve a disruption in the sea floor (which can be raised or dropped depending on the vector of the displacement). That motion of the sea floor gives the ocean water above it a shove - and since water is nearly incompressible, that energy gets transmitted outward as a seismic sea wave. Think of how a pebble dropped into water sends waves outward in a ring from the point at which the pebble hit the water - it’s pretty much the same phenomenon with tsunamis, although the disruption is coming from beneath the ocean surface.
The waves are detected in the Pacific Ocean through a network of sophisticated tidal buoys that transmit information back to data gathering centers. This data is used in conjunction with seismic data from the quake itself to estimate the starting point of the wave (above the sea floor which has been ruptured by the quake), and estimated times of arrival in relevant coastal regions. (In sum, there is no problem distinguishing a passing tsunami from a supertanker.)
For those commenting that a tsunami warning system is not a big deal and doesn’t need to be a large expense - I think you are seriously underestimating the level of effort and resources required. As an example, consider this list of requirements for a community to be considered “tsunami ready” by NOAA’s Pacific Tsunami Warning Center. Now consider that a lot of the infrastructure required to make the warning system work does not exist for the Indian Ocean region, since, as gouda pointed out, tsunamis haven’t previously been recorded in the Bay of Bengal. The cost of establishing and maintaining such a system would be a hard burden for countries in that region to bear (and to justify) for a natural hazard that occurs maybe once every few hundred years.
It is no secret to governments in the region that the USGS offers training programs, and certainly a wide variety of cooperative arrangements exist for bringing in outside scientists to assist with the assessment of seismic hazards. But again, it is a matter of priorities and limited resources.
By the way, how many posters here (not from Hawaii or Japan) would know that a rapid receding of the sea means that a tsunami is on the way? It’s easy to say in the aftermath that people should have known, or should have been educated about the hazard. However, I have no problem envisioning an enormous loss of life along the eastern US if something like the scenario in this OP took place in the summertime - because very few people here would know the warning signs, or be interested in learning out them, when we haven’t been hit by one in living memory. Everyone would flock along the shoreline to see what was going on, just as they did in Sri Lanka, before the tsunami arrived and took their lives.