Rizzo said that Soto was asymptomatic and that no one else on the roster was deemed ineligible following contact tracing.
Soto needs two negative tests before he can return to the field.
"We've got to make our plans to play without Juan for an extended period of time," Rizzo said on a conference call.
The Nationals did not include Soto on their initial 30-man roster for the 2020 season, instead placing him on the COVID-19 injured list.
"We're going to have to learn how to adjust to these things. The team that does the best job of keeping their star players on the field and healthy has a better chance of winning."
Soto, who had 34 homers and 110 RBIs last year, missed the first two weeks of "summer camp" for undisclosed reasons and didn't report until July 16. There is a Washington, D.C., rule mandating that anyone coming into contact with someone who had the coronavirus needed to self-quarantine for 14 days.
Soto, 21, was a breakout star of the Nationals' run to the 2019 World Series title, hitting .277 with five home runs and 14 RBIs in the playoffs.
Soto was replaced in left field by Andrew Stevenson in Thursday's lineup.
Martinez said anyone who was in close contact with Soto was tested early Thursday morning and will be tested again Friday. He said he's hoping the results will be available by Friday night.
ESPN's Jeff Passan contributed to this report.