Neither side could find a winner following the second half and 30 minutes of extra time, before the tournament hosts advanced with a penalty shootout victory.
"Today is the saddest day of my career," Isco posted on Instagram. "But, this is football and it doesn't stop, life doesn't stop.
"It's time to get up as we have always done and show pride in defending our country."
Spain's departure from the World Cup follows that of 2014 finalists Argentina and Germany.