Well, maybe. In the case of Saudi Arabia, the rulers are Sunni Moslem--sort of the 'liberals' of the Moslem crowd. However, the majority of inhabitants are NOT Sunni, and the royal family is divided. Some seem to be concerned that appearing to back the USA will lead to the overthrow of the S.A. gummint. Egypt is another quarter-stick waiting to explode; by no means is the Gummint stable, and it is FAR more Westernized than its population. Also recall that BinLadin and his #1 assistant are from Saudi Arabia and Egypt, respectively.
Rumsfield will have a hard time convincing these Gummints that it is in their best interest to support the USA and perhaps get overthrown as a result.