The major league home run and RBI leader is receiving fluids and is expected to be discharged from the hospital in 12 to 24 hours.
Rangers manager Ron Washington said Hamilton will sit out games Friday night and Saturday against the Houston Astros, and could be out Sunday, as well.
"We're going to miss Hamilton; that's the only feeling I have," Washington said. "We're going to move forward and continue to play baseball. You don't want your best player to not be in the lineup, but there's nothing you can do about it."
Washington said Hamilton was taken out late in Thursday's game against the Arizona Diamondbacks after complaining of a stomach ache.
Rangers officials believe Hamilton got the virus from his wife and children, who have been dealing with it recently.
"I did not [know he was hospitalized] ... I knew that his family was sick and that his stomach was bothering him a little bit [Thursday] night," Texas outfielder David Murphy said. "It's obviously tough when a player like that is not in the lineup, but again, we have plenty of depth and I'm sure we'll get by."
Added Michael Young: "It's part of playing in the big leagues. It's not always smooth sailing. You've got to find a way to keep fighting and grinding away. That's the nature of playing in the big leagues."
With Hamilton out, Texas called up 24-year-old Cuban outfielder Leonys Martin from Triple-A Round Rock. Martin started against the Astros, hitting ninth.
Texas placed right-handed pitcher Koji Uehara on the 15-day disabled list, retroactive to Sunday, to make room for Martin on the roster. Uehara has a right lat strain.
The move means the club will have to make another roster decision on Saturday when pitcher Justin Grimm is called up from Double-A to start against the Astros.
Hamilton missed three games against the Toronto Blue Jays earlier this season because of an upper respiratory illness. The issue persisted, and after hitting a big home run in helping the Rangers sweep the Blue Jays, Hamilton was given a day off on May 28 against the Seattle Mariners. He returned to the lineup the next day.
Hamilton is hitting .330 with 22 homers and 62 RBIs in 60 games this season. He was the AL player of the month in April and May, but has slowed so far in June, hitting .196 with one homer and five RBIs.
Martin was batting .344 with five homers, nine doubles and 22 RBIs in 31 games for Round Rock. He has a .414 on-base percentage and a .547 slugging percentage. He has hit safely with at least one RBI in seven straight games and is batting .394 with runners in scoring position. Martin has played mostly in center field.
Information from ESPNDallas.com contributor David Collier was used in this report.