Farnsworth said he was "shocked" by the decision and would appeal the penalty, which was to start Saturday night against Baltimore. He will be eligible to pitch until a hearing is held.
"I can't agree," he said. "We will appeal, and then we'll go from there."
Ramirez said Saturday that he doesn't think Farnsworth should have been suspended.
"He don't deserve that," Ramírez said, according to the Boston Globe. "He was just protecting his players. You know we hit [Alex Rodriguez] the game before."
Manager Joe Girardi also expressed his shock at the ruling by Bob Watson, vice president of on-field operations for Major League Baseball. The amount of the fine was not disclosed.
"I am shocked and I don't understand it. I disagree with it. It doesn't make any sense to me," Girardi said, adding that there have been far worse incidents that have gone without penalty this season.
Ramirez hit two home runs Thursday night in Boston's 7-5 win over New York. When he came up again in the seventh inning, Farnsworth threw a 97 mph pitch behind the star.
Ramirez dropped his bat, but never made a move toward the mound. Farnsworth wasn't ejected and plate umpire Larry Vanover immediately warned both dugouts. Ramirez eventually grounded out and the Red Sox never retaliated.
"The ball just slipped," Farnsworth said. "I didn't hit him. I'm not going to intentionally hit a guy in the head and maybe end his career.
"It wasn't my intention to hit him. I don't understand it. I didn't hit him. No one was ejected. So I don't know why he ruled the way he did."
A day before Farnsworth's close pitch, Red Sox reliever David Aardsma hit Rodriguez with a pitch.
Farnsworth has been suspended twice before in his career for throwing at hitters, in 2003 for three games and in 2005 for five games. Asked if he thought that was taken into consideration by Watson in making his decision, Farnsworth said he didn't think so.
No date was given for Farnsworth's appeal to be heard.
"The sooner we get it done, the better," Farnsworth said.
Information from The Associated Press was used in this report.