One who visits the sick on Shabbat should say, "Shabbat is not for petition, but may healing come swiftly." However, Rabbi Meir says, "perhaps it [Shabbat] will provide mercy."
We do not make personal petitions on Shabbat, leaving them for our weekday prayers. What then do we do when visiting the sick on Shabbat? During the week, prayer for healing is an element of our visit, but during Shabbat we should transcend our human needs so that we may gain a taste of the world to come. Nonetheless, we still remain in possession of our bodies on Shabbat and may still fall ill. Illness may sometimes inspire us to spiritual growth, but on the whole, most of us would say that we do not desire suffering or its rewards. So how do we approach the tension between the desire to overcome the physical on Shabbat and remain cognizant of the need for physical healing?
The first position above winks at the prohibition on praying for the sick, while then going on to explicitly pray for healing. This is the approach that the entire Jewish world has adopted in the Mi-Sheberach prayer for the sick on Shabbat. Indeed these very words, "Shabbat is not for petition, but may healing come swiftly," are included in that weekly prayer. I find Rabbi Meir's approach the more interesting one, however. His point is that we need not pray for the sick, because Shabbat itself is a time of mercy and favor before Heaven. Shabbat is its own prayer for all the needs of the people Israel.