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For a generalized Weyl Poisson algebra $A$, explicit sets of generators anddefining relations are presented for its Poisson enveloping algebra $\CU (A)$.Simplicity criteria are given for the algebra $\CU (A)$ and algebra of Poissondifferential operators $P\CD (A)$ on $A$. The Gelfand-Kirillov dimensions ofthe algebras $\CU (A)$ and $P\CD (A)$  are calculated. It is proven that the algebra $\CU (A)$ is a domain providedthat the coefficient ring $D$ of the generalized Weyl Poisson algebra $A$ is adomain of essentially finite type over a perfect field.  For the algebra $A$, the set of its minimal primes and the prime radical aredescribed and an equidimensionality criterion is given.  For the equidimensional algebra $A$ of essentially finite type, tworegularity criteria are presented.

The Poisson enveloping algebra and the algebra of Poisson differential operators of a generalized Weyl Poisson algebra