According to one commentary, many authorities agree that the flower mentioned by Solomon was some kind of bulb plant. Some Hebrew lexicons say this word is referring to a meadow saffron, many of which can still be found in the area today. In any case, there seems to be no reason why the rose, of which several varieties are common in Palestine, should not be meant by Solomon.
Gilead, part of ancient Israel's inheritance from God, gave its name to the balm of Gilead mentioned in the Bible (Genesis 37:25, Jeremiah 8:22, 46:11) as a healing ointment of great value, sometimes used as a trading commodity. There is no direct Biblical connection between the balm and the Rose of Sharon, but your question regarding any medicinal use of the plant was quite interesting since portions of today's roses are used for medical treatment.
The Bible does not use the rose as a symbol for Jesus but this phrase obviously has had tremendous resonance in modern culture.