Not quite true
Theory 1
The royal English rose is white and red. This rose was created to symbolise the marriage between Henry VII (House of Lancaster) and Elizabeth (House of York) and the end of the War of the Roses. Therefore, the ‘English’ rose is not red, but half red and half white. However, subsequent monarchs (such as Elizabeth I) continued to be associated with the red rose because they were descended from the House of Lancaster. So it is the red rose (rather than the red and white rose) which is often, incorrectly, seen as symbolising England or the English monarchy. Therefore the RFU might have selected the red rose for this reason.
Theory 2
Secondly, Lawrence Sheriff, the founder of Rugby School (where the game started) was presented with a coat of arms by Elizabeth I and in doing so she allowed him to use her red (Lancaster) rose [see above] on the crest. When he founded Rugby School the new institution based their own coat of arms on his, and so the red rose ended up on the Rugby School crest. The white kit worn by England was taken from the kit used at Rugby School, so it might be that a symbolic image was also taken from the school crest as a badge. Even more likely is the possibility that a combination of this connection with Rugby School and the incorrect identification of the red rose as the ‘English’ rose [see above].
Theory 3
The third possibility is that the red rose is actually used because it is also the symbol of the county of Lancashire. Of the subcommittee of men who selected the first England side in 1871, two came from clubs in Lancashire (Liverpool and Manchester). It is therefore possible that they were also responsible for selecting the England kit (including the red rose). However, it is highly unlikely that the other RFU Committee members would have accepted such a blatantly regional emblem on the national jersey and they would have reversed such a decision immediately.