Bostic was ejected from the game after further review.
Dalton is currently in the concussion protocol and has been ruled out of his team's road game against the Philadelphia Eagles on Sunday. Rookie Ben DiNucci will start at quarterback.
Washington head coach Ron Rivera discussed the hit with Pro Football Talk's Mike Florio after his team's 25-3 win over Dallas: "I told him, Jon, you're a veteran guy, you should know better. I didn't see it and I was really kind of asking everyone what happened. They said, 'He hit him, Coach.' And then when he got ejected, that really surprised me."
Rivera also said that he apologized to Cowboys head coach Mike McCarthy.
Bostic was the subject of widespread criticism following the hit, which some felt was worthy of a suspension:
"If this isn't a serious suspension, I don't know what is," Charles Robinson of Yahoo Sports wrote. "This is just a brutally unnecessary shot. And it has nothing to do with protecting a QB. You endanger any player's career with this kind of thing."
Football analyst Warren Sharp did not believe the penalty was severe enough in comments made after the fine news broke: "If this play only results in a $12,000 fine and no suspension, as medieval as it sounds, there isn't much disincentive for linebackers to continue KO'ing quarterbacks like this. Pretty bad look by the NFL that this is the only penalty for such a hit."
Bostic, an eight-year NFL veteran, has 52 tackles, four quarterback hits and two sacks in seven games this year. He had a career-high 105 tackles in 2019.
The Washington Football Team has a bye this week. Its next scheduled game is home against the New York Giants on Nov. 8.