This paper studies the Schr\"odinger operator with Morse potential on a right
half line [u, \infty) and determines the Weyl asymptotics of eigenvalues for
constant boundary conditions. It obtains information on zeros of the Whittaker
function $W_{\kappa, \mu}(x)$ for fixed real parameters $\kappa, x$, with x
positive, viewed as an entire function of the complex variable $\mu$. In this
case all zeros lie on the imaginary axis, with the exception, if $k<0$, of a
finite number of real zeros. We obtain an asymptotic formula for the number of
zeros of modulus at most T of form $N(T) = (2/\pi) T \log T + f(u) T + O(1)$.
Some parallels are noted with zeros of the Riemann zeta function.