The Wales international was sent off in the URC Round 15 game against Connacht at Rodney Parade on Saturday, 4 March 4 for a clumsy tackle on visiting flanker Conor Oliver in the 31st minute. Italian referee Andrea Piardi issued the red card after consulting with his TMO.
The ban could be reduced to two matches if Screech agrees to complete the World Rugby Coaching Intervention Programme, which automatically cuts a week off the sentence.
The Dragons are due to travel to meet the Ospreys on 25 March, head to Scotland for their European clash with the Warriors the following weekend and then head to high-flying Ulster on 14 April.
