Perhaps the question should be worded with the Elizabethan Age rather than Queen Elizabeth.
Certainly, the Elizabethans believed in the stars and the supernatural world and the cosmic order. So, the allusions to fate would and the "star-crossed lovers" would seem appropriate to them. Here is link that may prove interesting:
Yes, that is the case: fate was believed to control the lives and destiny of people in those days, but I don't see how this can be lined to Queen Elizabeth as a person. You might like to watch Shakespeare in Love, an excellent film that is innaccurate in lots of ways but offers a very interesting speculation about the involvement of Queen Elizabeth and Shakespeare.
I'm not sure that there is any direct connection between Shakespeare's play Romeo and Juliet and Queen Elizabeth I. It would be nice to imagine that she saw the play at court, but there is no record of such a performance:
It is possible to imagine all kinds of connections between the queen and the play (for example, that she may have sympathized with the young lovers' desire to marry because she had had to renounce marriage herself), and it's entirely likely that such connections have been argued. However, the fact that Elizabeth is barely mentioned in some of the best editions of the play (such as the Arden and Cambridge editions) suggests that any such arguments have not gained strong scholarly support.
The last history play written by Shakespeare is Henry VIII. It includes the birth and baptism of Elizabeth, the future Queen of England and the lofty speech delivered by Cranmer, a prophecy, on this occasion. Romeo is one of Shakespeare's earlier plays, the first tragedy of love. The play is sometimes interpreted as the encounter of Rome (and catholicism) and the Virgin Queen incarnated by Romeo and Juliet, the "fair sun". The letter "R", which is evoked by the nurse in act II, scene 3, is the initial of both Rosaline and Romeo's name but interestingly enough, it is also the initial of Regina, which corresponds to the signature of Queen Elizabeth.
what about the theme of fate? didnt people in that era also believe in destiny as it was portrayed in romeo and juliet?