A number of those visiting Whakaari when it erupted were from the cruise ship.
Friend of the couple Joan Mooney told the Ohio-based WKBN news the Ureys were out of intensive care and are expected to make a full recovery.
"They are making excellent progress. The doctors are amazed and their families are ecstatic because they've made such significant progress in a short time."
Despite receiving treatment in separate hospitals in the country, the newlyweds talked every day.
The Washington Post reported that Matthew's hands were so badly burnt he struggled to make a phone call to his family.
"It was very traumatic. A lot of tears, a lot of crying from Jan [Matt's mother], but she learned to kind of look for little victories each day," Mooney told WKBN news.
An update on the couple's hospital recovery was posted on their Gofundme page at Matthew's request.
"Matt wants everyone to know that he and Lauren are doing well," it read.
"They are hopeful that they will be returning home soon."
"Most of all, Matt wants to thank everyone for the support, encouragement, and for the donations that you have made to him and Lauren."