If circumstances so require, the blessing and exchange of rings can be omitted (Rite of Marriage #62) However, if it is retained in the ceremony, then both the bride and the bridegroom must give/receive the rings.
The words of the Rite give the best reason for both of them to do so. (Name), _____ take this ring as a sign of my love and fidelity. In the name of the Father and of the Son and of the Holy Spirit. If the rings are signs of love and fidelity should only one member of the couple pledge with a sign his/her commitment to this? Unequivocally, no.