Australia have struggled to recover from the crisis, with their men's teams losing a Test series to Pakistan in the United Arab Emirates and being whitewashed 5-0 by England in a one-day global series.
Amarfio, who joined the organisation in 2012, is the latest casualty of CA's cultural review which was commissioned following the ball-tampering scandal during the third Test against South Africa in Cape Town in March.
Roberts said his organisation would heed the review's call to meet with the Australian Cricketers' Association and mend its fraught relationship with the players' union.
Howard, who was in charge of CA's high-performance unit and meant to depart when his contract expired after the 2019 Ashes, will leave next week.
"Everyone at CA is focused on rebuilding and moving forward after what has been a turbulent year".
Appointed in 2011, former rugby worldwide Howard was a polarizing figure in the role, with former players and pundits questioning his lack of a significant cricket background.
However, CA announced on Wednesday that Howard is to depart next week, with former Australia global Belinda Clark taking on the role on an interim basis until a permanent replacement is named in 2019.
"We talk openly and honestly and obviously today's news was a shock as well", Carey told reporters in Adelaide.
"So I don't think it's a distraction ... it's all a bit of white noise for us".
Roberts, who was the chief negotiator in the MOU talks previous year, defended his appointment as chief executive, saying the review had commended him for his role in other initiatives such as uniting the states and territories, gender pay equity and commitment to community cricket.
Coach Darren Lehmann, chief executive James Sutherland, Peever and two other directors have left since the sandpaper scandal, while Steve Smith and David Warner were sacked as captain and vice-captain. Stephanie Beltrame, who was the head of CA's team during the recent broadcast rights negotiations, has been named as the interim EGM Broadcasting and Commercial.
Amarfio has long rubbed CA staff the wrong way, with the most notable example being a request that his secretary cook and serve him a hot breakfast in his glass-encased office.
The executive changes come on the heels of Chairman David Peever's decision to quit on Thursday after his position became untenable while long-serving board director Mark Taylor, the former Australia captain, announced his resignation on Monday.