Can non-Catholics pray the Rosary?

• The Apostles' Creed is ancient and straightforward.
• The Lord's Prayer is common to all Christians, I believe.
• The "Glory Be" should be unobjectionable for Trinitarian Christians.
• The first part of the Hail Mary is from Scripture (Luke).
• The insertion of the Name of Jesus in the prayer is natural.
• The second half is from theology. We have called Mary the "Mother of God" because of our belief that there is only one Person in Jesus, who is truly God the Son from before all time. Mary is the mother not of His divinity, but of his sacred humanity. But there is only on Person in Jesus, so she is His mother.
• To ask Mary to "pray for us sinners" is solidly based on the universality of sin among us believers. That we are all sinners in need of grace is common to all Christians, I believe. The thought that Mary is alive and capable of offering prayers for sinners is not a common Christian belief. This depends on your own convictions about the nature of the "communion of saints" (one of the articles of the creed). Only you can decide for yourself whether this is a stumbling block.
• Many people add the Fatima Prayer to the rosary.

"O my Jesus, forgive us our sins and save us from the fires of Hell. Lead all souls to Heaven, especially those in most need of Your mercy."

This is a great prayer theologically. I think you can say it without reservation, without thereby making any commitment to the reality of the Fatima apparitions.

• The traditional concluding prayer, the Salve Regina is only 1000 years old, so it does not go back to the time of the Fathers. You might, as a matter of charity to your patients, say it with them, but I imagine that its high praise of Mary might cause you some qualms.