The Mona Lisa vessel is owned by a Greek travel company and was flying a Bahamian flag. At present, five tugs, including a Dutch vessel, are being used to pull the 30,000-ton liner free from the sandbank where it ran aground on Sunday.
"We will not let the Mona Lisa leave our territorial waters until its owners pay for the rescue operation," Ivars Godmanis said.
The liner's passengers, primarily German pensioners, were evacuated on Monday during a five-hour operation.
The 173-meter (660-foot) vessel was heading for Riga with 984 people onboard when it hit an underwater sandbank 18 km (11 miles) off the coast of Latvia. Marine administration officials from the Baltic State have said the incident was caused by the "human factor".
The Mona Lisa was constructed in 1966 and has run aground nine times since then.
