It was initially thought Matthews had suffered from the usual road rash but the fracture in his left shoulder was discovered after further investigation.
"The diagnosis from the MRI scan is a non-displaced fracture of the greater tuberosity of the left upper arm," Team Sunweb's doctor Anko Boelens said.
"Because the fragment is not displaced, surgery, in this case, is not required. Michael will need to take some rest and afterwards undergo physiotherapy before restarting training on the rollers."
The team also said Matthews's Tirreno-Adriatico participation in mid-March may be in doubt.
"The healing process and Michael's pain levels will determine when he can return to the bike. After the weekend we will re-evaluate for his upcoming races and see if Tirreno is realistic," Boelens said.
Lennard Hofstede (NED) will replace Matthews at Strade Bianche.