The minister's remarks came weeks after a Jordanian teacher had called for allocating a section of the Al-Aqsa Mosque complex for Jewish worshippers.
Dawoud said his ministry had lodged a complaint against the teacher with the prosecutor-general's office.
"This edict harms the national interests of the Jordanian state," he told The Anadolu Agency. "It also harms Jordanian efforts to protect the Al-Aqsa Mosque and other religious sites in Jerusalem."
For Muslims, Al-Aqsa represents the world's third holiest site. Jews, for their part, refer to the area as the "Temple Mount," claiming it was the site of two Jewish temples in ancient times.
Israel occupied East Jerusalem during the 1967 Middle East War. It later annexed the holy city in 1980, claiming it as the capital of the self-proclaimed Jewish state in a move never recognized by the international community.
In September 2000, a visit to Al-Aqsa by controversial Israeli politician Ariel Sharon sparked what later became known as the "Second Intifada," a popular uprising against Israel's decades-long occupation in which thousands of Palestinians were killed.
Dawoud, for his part, called for a legal crackdown on anyone who posed threat to the Hashemite Kingdom's interests.
Weeks ago, a Jordanian teacher from the northwestern city of Irbid called for allocating a section of the Al-Aqsa Mosque compound for Jewish worshippers.
A video in which the teacher makes the suggestion has since gone viral, becoming fodder for local media.
On Sunday, Jordan's parliament passed a law calling for the punishment of anyone who issues religious edicts deemed harmful to Jordan's national interests.
Penalties laid down in the law include jail terms of up to three months and fines of up to $750.