Is it appropriate to leave a Catholic rosary in place on a cemetary headstone? Is it appropriate to put a rosary on a relatives headstone and leave it there even if it starts to discolor and rust the headstone?  Would it be a desecration of the grave if someone removed it because of possible negative effects?
I don't believe that there is any formal Catholic doctrine on the use of rosary beads (technically "a rosary" refers to the prayer, or rather sequence of prayers, not to the physical beads themselves). While the Catechism of the Catholic Church mentions the rosary (always referring to the prayers), it contains no description of how the beads ought to be treated. At most, it simply says in paragraph 1676:


  Pastoral discernment is needed to sustain and support popular piety [including such devotions as saying the rosary or using the rosary beads for a pious purpose] and, if necessary, to purify and correct the religious sense which underlies these devotions so that the faithful may advance in knowledge of the mystery of Christ.


If the beads are blessed, as they often are, then of course they ought to be treated with reverence.