Lambeth Palace said at the weekend that, "in light of the sheer volume of requests the Archbishop receives, and the many pressures on his time and resources", he had reluctantly declined the position, which his four predecessors had accepted.
A spokeswoman said that the Archbishop, nevertheless, had "enormous admiration for the RSPCA".
The director of the Oxford Centre for Animal Ethics, the Revd Professor Andrew Linzey, said that the refusal was "incomprehensible and regrettable".
"When I was a member of the RSPCA council in 1976," he said, "I first proposed Archbishop Coggan for the presidency of the Society, which he accepted. The role was changed to vice-patron by Archbishop Carey, but, for four decades, successive Archbishops have had an honorary role within the Society.
"It is very difficult to know why the Archbishop would want to distance himself from an organisation specifically founded as a 'Christian society based on Christian principles' by an Anglican priest, Arthur Broome, in 1824.
"Invitations to church dignitaries to become patrons of philanthropic societies and institutions are not personal honours to the individuals invited, but honours to the Church. The C of E should welcome and celebrate the fact that so many bodies still want contact with the Church at all."
In a letter to the Church Times, the Rt Revd Dominic Walker, who retired as Bishop of Monmouth a few weeks ago, said that he was "disappointed" that Archbishop Welby had turned down the post.
Bishop Walker is president of the Anglican Society for the Welfare of Animals, and an honorary vice-president of the RSPCA. "I hope that the decision will not give the general public a false message that the Church is not concerned about animal welfare."
The RSPCA said that it "respected" the Archbishop's decision.
Pro-hunt campaigners welcomed the decision, saying that the
"politics" of the RSPCA might have informed the Archbishop's
decision.The head of campaigns at the Countryside Alliance, Tim Bonner, said: "The RSPCA maintains that its increasingly political agenda is having no effect on the organisation, but the loss of a high-profile vice-patron suggests this is simply not the case."
A Sunday newspaper reported that Archbishop Welby had formally attended a live shoot, and also went clay pigeon shooting. It quoted an estate manager from the time who recalled that the Archbishop was a "reasonable shot and a nice bloke".
Lambeth Palace said in a statement: "The Archbishop went on a live pheasant shoot on one occasion in the mid-1980s while working for Enterprise Oil. It held no interest for him and he never went again. This has no bearing on his recent decision not to patron the RSPCA, which is a decision he made regretfully due to time constraints."