Dalton, who was making his second start in place of injured Dak Prescott, took a hit to the head from Bostic as he slid to the turf with 6:22 to play in the third quarter.
According to a team spokesman, Dalton was alert and doing much better than he was when he left the field after the hit.
Cowboys coach Mike McCarthy said he was not pleased with the passive reaction teammates had to the Bostic hit.
"We speak all the time about playing for one another, protecting one another," McCarthy said. "It definitely was not the response you would expect."
Washington coach Ron Rivera told ProFootballTalk.com that he apologized to McCarthy for Bostic's hit.
"I went up to him and apologized to him for it," Rivera told the website. "I wanted him to know that we don't play that way, and it's unfortunate that it happened."
ESPN's Todd Archer contributed to this report.