The warning came during an address to more than 300 church-goers in the county.
The retired Anglican bishop told clergymen and worshippers it was "urgent" that more was done to bring young people into the church so that its very existence was not put at risk.
The 78-year-old was making a keynote speech at Holy Trinity Church in Meole Brace, Shrewsbury, as part of the Shropshire Churches Conference 2013.
He admitted it was a challenge to get new people, particularly young people, into churches and claimed scores of ministers up and down the country were struggling under the "heavy burden of defeat".
But he added: "I am convinced that churches can grow, must grow and should grow. But to sit in a cold church, looking at the back of people's heads, is perhaps not considered the most exciting place to meet new people and hear prophetic words.
"So we have a problem getting people to church because it is not something that is natural to people in their lives nowadays.
"There is a prevalent view that people don't want to hear what we have to say any more. How can we say that?
"There is so much violence, too many divided families, too little job security, too many young people with nothing to aim for.
"It is still the case that people are essentially looking for spiritual fulfilment."
Lord Carey said psychologists had claimed there was a clear link between faith and better mental health.
He pointed to the success of foodbanks and credit unions as examples of what churches could do when thinking outside the box.
But he warned: "One of the most worrying, most urgent groups we need to invest in is young people. We ought to be ashamed of ourselves.
"We are one generation away from extinction – if we do not invest in young people there is going to be no-one in the future."
See also: Women bishops proposal at Synod