Terry was cleared of racially abusing Anton Ferdinand by a court on Friday but Taylor says the repercussions of the case will be lasting.
Taylor, chief executive of the Professional Footballers' Association, wants the FA's Respect campaign to be stepped up for the new season with players taking more responsibility for being role models.
He also suggested the FA are in "a difficult position" over whether they charge Terry over the same incident.
Taylor told the Press Association: "You have to respect what has happened in court and there has been a judgement made, but it has not been the best of weeks for football.
"It has been an unedifying process and the game has been damaged as a result of the dirty linen being washed in public.
"I now want to see an improvement in the Respect campaign. It is not just the line between what is banter and what is illegal, so much of what we have heard this week needs to be cut out.
"The players are role models and everything now gets picked up in a match on television. These insults and this language leads to things that should not be said."
Liverpool's Luis Suarez was banned by the FA for eight matches for racially abusing Patrice Evra last season, and Taylor says players should be in no doubt about a line that cannot be crossed.
"People will say it has set back a lot of initiatives and it has not been comfortable for the game for us to have had two of our members in that situation," he added.
"There are no grey areas any more. Players now know exactly what they can say and what they can't. It's the biggest game in the world and players who earn big money know the cameras and microphones are always on them and should be very mindful of what they are saying and doing.
"The Respect campaign has got to be stepped up."
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In terms of a possible FA charge, the body will spend next week studying the findings of the Terry verdict before making a decision.
Taylor said on that issue: "Football has shown that it has been quick to deal with such issues and move on, and needing a lower burden of proof [than court] puts the FA in a difficult position now.
"Some people may say we should we draw a line under it and really reinforce the campaign at the start of the new season, but at the same time we must not sweep it under the carpet and say everything is fine when it's not."
PFA chairman Clarke Carlisle claims trying to eradicate the sort of obscene language highlighted by the Terry case would cause "mayhem", but insists it can be done.
"I think generally there is a very high level of abusive language that goes on on the pitch and it just seems to be par for the course, but I don't agree that it should be that way," Carlisle told Radio 5 Live.
"In order to change it it would take a very strong line from the referees, a very strong line from the governing bodies and it would probably cause mayhem for the short-term period.
"We do have the regulations within the game in order to stamp out any kind of foul and abusive language, but they are just not enforced to the nth degree.
"And I think if we were to adopt a line of social responsibility, because sport is so influential, especially football that's watched by hundreds of millions on a weekly basis, it would create a short-term period of mayhem.
"But I think if players were being sent off and banned because of the language that they are using then it would cause them to address their behaviour and it would cause the clubs to come down on it too."