No, there is no capitalize when referring to rosary beads. You may choose to capitalize when referring to set of prayers called the Rosary, but not for rosary beads.
Rosary
I believe they are called a rosary.
By the prayers of the Holy Rosary.
The Rosary was developed as an aid to support the faithful in their prayers. Specific prayers are not so much assigned to each bead as there is a bead to mark each prayer.
Rosary beads are used in prayer by Catholics to keep track of the prayers they are saying. Each bead represents a specific prayer, such as the Our Father or Hail Mary. The person holding the rosary beads moves from one bead to the next while reciting the prayers. This helps them focus on their prayers and meditate on the mysteries of the rosary.
The Roman Catholic Church and the Orthodox Church are the two main religious institutions that pray the rosary, though other Christian denominations and even other religions pray the rosary.
Roman Catholic AnswerThe link below has the prayers of the Rosary in Latin and in English.
Dominic de Guzman did not 'invent' the rosary but did promote it.
A typical rosary consists of 50 Hail Mary prayers.
The Rosary is a string of prayers so since you say "at a catholic rosary" I assume you mean this is a public event where the Rosary is said. There will probably be one person (probably a priest) leading the prayers and everyone else joining in for the last few lines of each prayer, like a chant. It could also be in latin.
It's the other way around, the physical rosary was put together the way it was so that you would be able to see where you are in your praying of the Rosary.