According to the October 1, Fiscal Year 2007, Airline City Pair Awards spreadsheet, yes. I opened it and did a search for DXB and the only result was NYC-DXB on AA out of JFK/EWR/LGA.

All this means is that travel between NYC and DXB should be on AA or an AA codeshare. Since there are no other GSA city-pair awards, any other origination city could be served by any airline, with preference going to US-flagged carriers (and foreign carriers operating on a US carrier's codeshare).

I'm not familiar with the route, since I usually fly to BAH (UA to LHR, BA (AA codeshare?) to BAH). However, doing a dummy booking SAN-BAH through the DTS travel portal I could pick flights on Northwest/KLM, UA/LH, DL, CO/AF.