Admiral Richard E. Byrd retired from the US Navy as rear admiral in
1930.  My uncle recorded in his diary that "Admiral Byrd" inspected
his military unit on 9 June 1942 on Tonga in the South Pacific, during
World War II.  I cannot find what Admiral Byrd did between his 1939
and 1946 expeditions.  Perhaps he performed ceremonial duties for the
US military.

Was the Admiral Byrd who inspected my uncles unit retired Rear Admiral
Richard E. Byrd, or was it another Admiral Byrd?

Dear fulks,

Though I found no explicit mention of Tonga, there can be no doubt
that the "Admiral Byrd" who inspected your uncle's unit, was indeed
famous Richard E. Byrd.

After he had returned from the U.S. Antarctic Service expedition on 5
May 1942, Admiral returned to active service and was re-assigned to
the Navy Bureau of Aeronautics. As such, he was the senior member of
the South Pacific Advanced Base Inspection Board which toured bases in
the south Pacific in May and June 1942.
It appears self-evident that Admiral Byrd and his staff during their
tour also inspected Tonga, because the island was important for
keeping open a line of supply between Australia / New Zealand and the
USA.

Since I was not able to find another "Admiral Byrd" who toured and
inspected the South Pacific island bases in late spring and early
summer 1942 (which would be, by the way, far too much coinidence), it
is undoubtable that Richard Evelyn Byrd, the Antarctic explorer, was
the Admiral who was on Tonga on 9 June 1942.

Sources:

Naval Historical Center: Papers of Rear Admiral Richard E. Byrd,
1941-1945
http://www.history.navy.mil/ar/bravo/byrd.htm

Ohio State University : Admiral Richard E. Byrd, 1888-1957
http://www-bprc.mps.ohio-state.edu/AboutByrd/AboutByrd.html

South-Pole.com: Antarctic Explorers - Richard E. Byrd, Byrd Antarctic
Expedition III
http://www.south-pole.com/p0000109.htm

Ohio State University : Chronology of Admiral Richard E. Byrd
http://www.lib.ohio-state.edu/arvweb/polar/byrd/byrdchrono.htm

American Samoa: The Samoan Historical Calendar, 1606-2001
http://www.asg-gov.com/historicalcalendar_may.htm

Search terms used:
"Richard E. Byrd" 1942 pacific
://www.google.de/search?hl=de&ie=UTF-8&oe=UTF-8&newwindow=1&q=%22Richard+E.+Byrd%22+1942+pacific&meta=
"Richard Byrd" admiral 
://www.google.de/search?hl=de&ie=UTF-8&oe=UTF-8&newwindow=1&q=%22Richard+Byrd%22+admiral+&meta=
tonga byrd admiral 1942
://www.google.de/search?hl=de&ie=UTF-8&oe=UTF-8&newwindow=1&q=tonga+byrd+admiral+1942&meta=

Hope this helps!
Best regards,
Scriptor

The research and response are exactly what I wanted and expected.  The
only additional possibility would be some kind of interesting anecdote
about the situation, but I assume that Scriptor did not encounter any
such anecdote.  Tom Fulks