The England Under-21 international made a surprise move to Italian side Sampdoria this week, despite having made 60 league appearances for the Championship club in two years.
Radrizzani admitted that manager Marcelo Bielsa was "not happy" about Vieira's departure, believed to have yielded Leeds up to £10m, but stated that "as a custodian of the club I need to be rational".
Speaking to BBC Sport, the Italian said: "The investment we've made in the past year does not allow us to do anything else to find more resources.
"The sacrifice was very hard for me to do because he's a great player. He was one of best players to come from our youth but I cannot put the club in danger in the future.
"We thought it would be a sacrifice we could take, and still maintain the team and invest where we need more. For example, we have brought in all the players the manager asked for, and perhaps we can look for one or two more this summer."