At the level of the newly enlisted man, the answer was generally no, as the needs of the military often pre-empted most individual choices even in cases where volunteering was nominally possible.

The lion's share of the Army's effort was in Europe, and the vast majority of the Navy and Marine Corps spent the war or portions of the war in the Pacific. Assignment to a particular theater was generally dependent upon higher unit assignment, which was determined by the needs of the military itself. Aptitude in a specific task could result in assignment to a particular type of unit, and certain types of units were predominantly deployed to only certain theaters, resulting in a higher likelihood of being assigned to the theater in question.

Voluntary enlistment was mostly discontinued on December 5, 1942 via Executive Order 9279

30 April 1945 Reported Actual Strengths in Principal Theaters

30 April 1945	ETO	MTO	ETO+MTO	SWPA	POA	CBI	Combined	Total Army
Divisions	819,342	102,485	921,827	183,798	80,834		1,186,459
Non-divisional Combat Troops	595,418	37,797	633,215	113,318	102,076	15,766	864,375
Total Ground Combat Troops	1,414,660	140,282	1,555,042	297,118	182,910	15,766	2,050,835
Non-divisional Service Troops	828,726	128,307	957,033	214,835	142,579	69,898	1,384,345
Air Service	439,425	153,005	592,430	173,343	75,438	90,949	932,160
Replacements	171,933	34,551	206,484	34,203	20,371	4,355	265,423
Patients in SOS Hospitals	128,305	15,679	143,984	41,135	6,423	1,439	192,981
Unclassified	27,598	4,972	32,570	29,500	19,851	10,447	131,636
Total	3,065,505	493,876	3,559,381	802,504	450,952	198,830	5,011,657	8,290,993

Interesting situations could result where, when during the activation process of a unit, numbers of men who voluntarily enlisted or were drafted into the Army at a certain time from the same region could end up being assigned to the same unit and eventually serving in the same theater. To economize activation or expansion of new units, the Army would often send numbers of fillers from Service Command reception centers near where a unit was to begin training.

Service Command	Headquarters	States
1	Boston, MA	CT, ME, MA, NH, RI, VT
2	Governors Island, NY	DE, NJ, NY
3	Baltimore, MD	DC, MD, VA, PA
4	Atlanta, GA	AL, FL, GA, MS, NC, SC, TN
5	Columbus, OH	IN, OH, KY, WV
6	Chicago, IL	IL, MI, WI
7	Omaha, NE	CO, IA, KS, MN, MO, NE, ND, SD, WY
8	Dallas, TX	AR, LA, NM, OK, TX
9	Fort Douglas, UT	AZ, CA, ID, MT, NV, OR, UT, WA

The 35th Infantry Division (National Guard from Kansas, Nebraska, and Missouri) received most of its fillers from the Seventh Service Command in early 1941 after it moved to Camp Joseph T. Robinson, Arkansas, and this is still in evidence in its subordinate units over three years later in May 1944 as the division prepared to ship to Europe. The original "2072" men (federalized Nebraska National Guard personnel) were swimming in a proverbial sea of "17" and "37" men and men from other Service Commands, soon to be diluted even more as loss replacements drafted from all over the country flowed in beginning in mid-July 1944.

The assignment of units to theaters also depended upon when the unit was activated, when it had completed its training, where it had done such training, and the overall strategic situation.

Men assigned as loss replacements, the main focus in training in the ground and air Army after 1943, constituted about two-thirds of newly-inducted Army personnel by the summer of 1944. As most replacement training centers were in the lower Great Plains or southeastern United States to facilitate year-round training, they were filled with men from all over the country.

I speak at significant length about the acquisition and training of manpower for the U.S. Army during World War II in these other answers;

• How were elite divisions in ww2 such as the 101st airborne from Band of Brothers able to have replacements often even though training took 2 years?

• Were criminals allowed to be part of the US army in WW2?

• Why were injured soldiers that had recovered sent back to different units in WW2?

• The utilization of older men by the Army, particularly in relation to the manpower troubles of 1943-1945

• It's 1943 in America and I just got drafted. Do I have any say in what job I do or what branch I serve in? Is my situation different if I am black vs white?

• Conscripted men in the US army generally aren't portrayed as any different from volunteer soldiers while in other countries they are shown as poorly trained, poorly armed, and with very low morale. Were conscripted units in America trained, armed and treated just as well as its enlisted men?

• Training of infantry replacements

• General mobilization and training of infantry divisions II

• Did US troops in WW2 experience continuity of unit integrity as they progressed from basic training to combat?

• [American manpower situation during WWII] (https://www.reddit.com/r/AskHistorians/comments/4yrj2r/ww2_ive_heard_contradicting_reports_on_how/?)

• General mobilization and training of infantry divisions

Sources:

Palmer, Robert W. The Army Ground Forces: The Mobilization of the Ground Army, Study no. 4. Washington: Historical Section, Army Ground Forces, 1946.

As howling cow notes, the Marines fought exclusively in the Pacific (well almost exclusively, as I think some individual Marines served as liasons or observers or in the OSS and such in Europe, but no actual combat units that I'm aware of). Before voluntary enlistment ended, someone who wanted to fight the Japanese could enlist in the Marines and thereby essentially choose the Pacific theater. I seem to recall reading memoirs where this conscious choice was made, although I couldn't give you a name now.