"We want to inform you that Linas Balsys is currently under the care of doctors and his family. We sincerely thank the medical personnel and all those who actually offer their real help and support, not just on the TV screen," the family said in a statement.
The statement asks the media "not to escalate and not to harass" and calls on politicians not to use the situation "to advertise themselves, especially through obvious slander and lies".
Balsys, the leader of the Lithuanian Green Party, last Thursday agreed to blow into a breathalyzer device when asked to by journalists, producing a reading of 1.82 per mille of alcohol, and left the Seimas building shortly afterwards.
The VIP Protection Department reported the suspected drunk-driving incident to the police. It was said several hours later that Balsys was at home but refused to speak to police officers.
The police said they would check surveillance video records from the Seimas and public CCTV cameras in the capital and would consider bringing the parliamentarian to administrative liability.
Balsys, a former advisor to President Dalia Grybauskaitė and a journalist, is serving his second term as lawmaker.