Premiership Rugby's decision followed new guidance from the United Kingdom government on Monday.
British prime minister Boris Johnson encouraged people to avoid mass gatherings and all "non-essential" travel and contact with others.
"The safety of our fans, players and staff is our first priority, and we would like to wish all of those who are affected our heartfelt best wishes and a very speedy recovery," a Premiership Rugby statement read.
"We'll continue to work closely with DCMS [the Departure for Digital, Culture, Media and Sport] and Public Health England, and to follow the advice of medical experts, and work with our clubs to help them to support their own wider communities at this time."
The decision to call off the European rugby season was taken during a conference call on Monday.
It means the matches in the Champions Cup and Challenge Cup scheduled for April 3, 4 and 5 will not now go ahead as planned, although organisers hope to rearrange the fixtures for later dates.
In a statement, the board of European Professional Club Rugby (EPCR) said: "While respecting all further directives by governments and local authorities, EPCR, in conjunction with the relevant leagues and unions, remains committed to trying to find a solution which will enable it to complete the 2019-20 Champions Cup and Challenge Cup tournaments at such a time as that aligns with preventative measures undertaken by an overwhelming majority of sport's governing bodies and tournament organisers in Europe.
"EPCR's priority is the health and welfare of players and club staff, and an update regarding the knockout stages of the tournaments will be communicated to all stakeholders as soon as practicable."