Is ongoing economic growth possible without additional non-renewable inputs? Can ongoing economic growth occur without use of additional non-renewable inputs? That is, any additional inputs would only be from renewable resources: all non-renewable resources would be reused, recycled or removed from the economy.
I don't think anyone knows how to "make" an economy grow, but there's a simple channel through which economic growth can occur under fixed non-renewable inputs: innovation. 

Economic growth does not require increasing use of inputs; if there's profit to be gained through more efficient use of fixed quantities of inputs, or by increasing output of renewable resources, people will have an incentive to figure out how to do so.

Regarding whether fixing a minimum price of a nonrenewable resource is a good idea: possibly, but likely no. Typically, the most efficient ways to reduce the use of a particular input are to either charge the marginal social cost of the use of that input as a tax (allowing quantities to vary freely, which you disallow in your scenario), which incidentally has the effect of setting a minimum price but which requires that you know the marginal social cost of the use of that resource, or to do cap-and-trade, where you fix the quantity available and let market prices determine the most efficient allocation. Setting a minimum price in a cap-and-trade scenario can result in inefficient underuse of the resource.

If you're interested in the theory of the price of nonrenewable resources, you might want to start with Hotelling's Rule.