The letter dated today comes amid debate about a possible exemption for Catholic adoption agencies from a new law regarding alleged discrimination against homosexuals.
The Equality Act 2006, which comes into force in April, bans discrimination in the provision of goods, facilities and services on the basis of sexual orientation.
A possible compromise to the law was proposed, under which Catholic agencies could refuse to accept homosexual couples, but would have a duty to refer them to agencies that would accept them.
However, the Lord Chancellor, Lord Charles Falconer, told BBC television: "We have introduced laws which prevent discrimination against people on the basis of their sexual orientation; those laws should be given full effect.
"We do take the view in this country that you shouldn't be discriminated against on that basis and think that applies to everybody, whatever your religion."
To act against principles
Cardinal Murhpy-O'Connor, 74, clarified that the Church "utterly condemns all forms of unjust discrimination, violence, harassment or abuse directed against people who are homosexual," and "teaches that they must be accepted with respect, compassion and sensitivity."
But he added that "to oblige our agencies in law to consider adoption applications from homosexual couples as potential adoptive parents would require them to act against the principles of Catholic teaching."
Cardinal Murphy-O'Connor explained that Catholic agencies already act in accordance with the proposed exemption to the law.
"Homosexual couples are referred to other agencies where their adoption application may be considered. This 'sign-posting' responsibility is taken very seriously by all Catholic adoption agencies," the cardinal said.
After citing the "excellent track record" of Catholic adoption agencies, the prelate said "it would be an unnecessary tragedy if legislation forced the closure of these adoption services."
"This is an appeal for 'fair play,'" the cardinal added, "particularly for those many children, Catholic or not, who continue to benefit from the widely recognized, professional and committed adoption services provided through our Catholic adoption agencies."