La Liga authorities made the decision after Real Madrid confirmed its senior football team is in quarantine after one of the club's basketball players tested positive for the virus.
Real Madrid's football and basketball players share facilities at Ciudad Real Madrid, the club's training ground in Valdebebas.
A club statement read: “A player from our basketball first team has tested positive for the COVID-19 coronavirus.
“Since then, the recommendation has been made to quarantine both the basketball first team and the football first team, given that the two squads share facilities in Ciudad Real Madrid.
“Likewise, it has been decided to close the facilities at our training ground and it is also recommended that all Real Madrid personnel who work there remain in quarantine.”
La Liga has also followed Italy's Serie A in suspending its matches, after a meeting on Thursday with the Spanish Football Federation and the Spanish Players Union (AFE).
It had been decided on Tuesday that games in Spain's top two divisions would be held behind closed doors but those fixtures over the next 14 days will now be suspended.
Real's second leg away to Manchester City in the Champions League last 16, scheduled for next Tuesday, is also in serious doubt.
Barcelona have already confirmed their tie against Napoli at Camp Nou on Wednesday will be played without fans, although UEFA is facing mounting pressure to postpone completely both the Champions League and Europa League.
Domestic league matches in France, Germany and Portugal have been put behind closed doors while Serie A in Italy has been suspended until April 3.
A La Liga statement read: “In view of the circumstances known this morning, referring to the quarantine established in Real Madrid and the possible positive tests from players at other clubs, La Liga considers that the circumstances now exist to continue with the next phase of action against COVID-19.”
“La Liga agrees to suspend at least the next two matchdays,” the statement added. “This decision will be re-evaluated after the quarantine at affected clubs and other possible situations that could occur have ended.”
Spain has reported 2,968 coronavirus cases and 84 deaths, according to figures on Thursday. Madrid is by far the country's worst-hit region, with 1,388 cases and 38 deaths.
Casos de #Coronavirus en España, actualizados a las 13h de hoy:
País Vasco 346
La Rioja 205
TOTAL 2968 pic.twitter.com/w9JIjKcIPe
— Salud Pública (@SaludPublicaEs) March 12, 2020