The incident occurred in the Sept. 28 match between the Championship sides, where the FA concluded on Friday that the Spaniard's remarks "made reference to race and/or colour and/or ethnic origin" of Leko.
The ruling was made by an independent regulatory commission, which will not publish their reasons for their decision until next week.
The FA also instructed Casilla to attend an education course.
Casilla posted on Twitter after the ruling, where the former Espanyol and Real Madrid keeper said: "[The] last five months have been the most difficult ones in my professional career.
"I am sure my family, friends, colleagues and the different technical teams whom I have been working with for all these years know that I would never use my words with a racist meaning.
"I would like to deeply thank Leeds for its support during this time and I appreciate the support of our amazing fans, whose warmth has strengthened me."
In a statement, Leeds said: "We would like to make it clear that we do not tolerate any form of discrimination within our football club and we are a leader in the fight against discrimination within our wider community. However, it is important to recognise that Kiko has always denied making any racist comment.
"The FA panel have based their decision on the balance of probability rather than proving Kiko to be guilty beyond reasonable doubt, which we have always believed is the more appropriate burden of proof."
The suspension of Casilla comes when Marcelo Bielsa's side remain in the hunt for the Championship title and automatic promotion to the Premier League. Leeds are in second place in the table and five points clear of third-placed Fulham.
Leeds play Hull City away on Saturday. After missing that match, Casilla will sit out fixtures with Huddersfield Town, Cardiff City, Fulham, Luton Town, Blackburn Rovers, Stoke City and Swansea City before being eligible to return against Barnsley on April 18.