Found some interesting stuff (other than Larry Bird). Here is a sample that might answer your question:

WHERE DID IT GET ITS NAME? French Lick was named by French settlers traveling through the area for the abundant water supply and the benefits derived by their livestock from licking the mineral deposits.

H I S T O R Y: The site's first building was a government fort built in 1812. In 1834, renowned physician Dr. William Bowles purchased the site and three mineral springs from the State of Indiana. Sensing the draw of the natural beauty and the commercial potential of the mineral waters, Dr. Bowles opened the first hotel within a few years. Both the hotel and the water from its most famous well, sold nationally as "Pluto Water", became very popular. After the original wooden structure burned in 1897, the site was purchased in 1901 by Tom Taggart, former mayor of Indianapolis and leader in the National Democratic Party. The present hotel opened in the early 1900s and quickly became synonymous with luxury and relaxation, a continuing draw for the country's rich, famous and politically influential. French Lick was also a gambling mecca, the site of many opulent casinos and the unofficial retreat for the Democratic Party. In 1931 at the National Governor's Conference, Franklin Roosevelt gained support for the presidential nomination.

After World War II, the grand hotel passed through numerous hands. In 1991, the property was purchased by the Luther James Family of Louisville, Kentucky, and extensive renovation was accomplished. In 1997, the Boykin Management Co. from Cleveland, OH, bought the resort and is continuing to restore the elegance, grandeur and luxury to this beautiful and historic property. The Boykin group has been in the hospitality industry for over 25 years and manages 26 hotels throughout the country.

I wouldn't be surprised to find Sheraton among its many owners.