The US military operates two constellations of such satellites, whose purpose is to detect and provide early warning for enemy ICBM launches using infra-red detection sensors. The older constellation is the DSP (Defense Support Program) series of satellites in geostationary orbit. There is also the newer SBIRS (Space-Based Infrared System) constellation, consisting of one geostationary satellite (SBIRS Geo-1) and two SBIRS sensors piggybacked on HEO satellites (USA 184 and USA 200).
We know that the DSP satellites have, in the past, frequently observed meteoric fireballs. It is therefore widely believed (and indeed likely) that the system should have been able to detect the UARS reentry fireballs as well. The problem is that post-9/11 the DoD has stopped declassifying meteor detections (which were previously shared with meteor scientists). Which makes you wonder whether, if they did detect the UARS reentry fireball, they would be forthcoming with that information. Probably not.
Would the UARS reentry have been visible from one of the DSP or SBIRS satellites? Would they cover the relevant areas? Yes they would.
Below map shows the location (for 4:16 UTC [edit 28 Sep: this was written before the reentry time was revised to 4:00 UTC, see here]) of UARS plus it's track, and the locations of the relevant satellites.
Yellow dots are the block 5 DSP satellites, white dots the SBIRS satellites. The green circle outlines show the coverage area of DSP F16, DSP F20, and SBIRS Geo-1.
The DSP's and SBIRS GEO-1 are geostationary and hence always above the geographic spot depicted in the map (with some minor latitudinal variation): for the HEO SBIRS platforms USA 184 and USA 200 the position plotted is for 4:16 UTC.At that time USA 184 was near apogee and basically almost in the same position (in geographic subsatellite point terms) for an hour on each side of 4:16 UTC. USA 200 was moving towards perigee, but would have UARS in view during the whole Africa pass of the latter.
DSP F16, DSP F20, SBIRS GEO-1 over the eastern Pacific as well as the SBIRS platform USA 184 over Siberia would cover the approach track over the Pacific and nominal center of the reentry window of UARS. Basically, they cover UARS on it's final track from New Zealand to over Canada.
Beyond Canada (would UARS have survived well beyond 4:16 UTC), DSP F17 over Brasil and the SBIRS platform USA 200 moving over Africa would have taken over, joined by DSP F18 plus DSP F21 and DSP F22 (all over Africa or the Indian Ocean) once over Africa.
I also checked whether the experimental satellites in the STSS (Space Tracking and Surveillance System) series would have been able to capture it: turns out they would not have, as these satellites (in low earth orbits) were not near the relevant part of the UARS track at that time.
In conclusion: there is plenty of possibilities for the US military to have detected the UARS reentry from space, using their space-based assets (DSP and SBIRS) in GEO and HEO. Even if groundbased tracking facilities were sparse over UARS' final track, the space-based sensors should have been able to observe and pinpoint the reentry.
Yet, I suspect that if these observations exist (allowing the DoD a clear indication as to where UARS debris might have showered down), this information will not be released to the public.
(text slightly editted 27 Sep to clarify USA 200 movement)