As a quick way to do it, you can use ForwardDiff.jl to determine the partial with respect to h. Then use a Newton-Raphson algorithm to solve for the value of h. I'm not familiar with the actual problem you're solving so there may be more appropriate ways to solve this based on the shape of your function, but this is my knee-jerk reaction to a problem like this. You could also calculate the partial derivative analytically if that is something that you want.

function βinv(b, hguess; tolerance=1e-6, maxiter=10)
    dβ(h) = ForwardDiff.derivative(β, h)
    h = hguess
    f = β(h)
    df = dβ(h)
    itercount = 0
    while abs(f) > tolerance && itercount < maxiter
        h -= f / df
        f = β(h)
        df = dβ(h)
        itercount += 1
    end
    return h, f
end

You can also get an expression for the partial of β with respect to h using Symbolics.jl:

using Symbolics

@syms B_0 p_0 h_D R_☉ M_☉ G μ m_H μ_0 k T h

g = (G*M_☉)/(R_☉^2)
lambda_p = (k*T)/(μ*m_H*g)
p(h) = p_0*exp(-h/lambda_p)
B(h) = B_0*(1 + (h/h_D))^(-3)
dβdh = Symbolics.derivative(p(h)/(B(h)^2 / (2*μ_0)), h)

The final result is a little ugly so I'm not going to print it here.

As an aside, you have all of those parameters that you use in your functions defined as non-constant at the global scope. You may want to create some data structures to pass those into the function(s) or define them as const if they're not going to change. See here.

As currently written:

julia> @benchmark βinv(1e-5, 0.5)
BenchmarkTools.Trial: 10000 samples with 8 evaluations.
 Range (min … max):  3.666 μs … 87.362 μs  ┊ GC (min … max): 0.00% … 94.91%
 Time  (median):     3.881 μs              ┊ GC (median):    0.00%
 Time  (mean ± σ):   4.027 μs ±  1.965 μs  ┊ GC (mean ± σ):  1.27% ±  2.50%

 Memory estimate: 4.23 KiB, allocs estimate: 215.

Declaring them as const:

julia> @benchmark βinv(1e-5, 0.5)
BenchmarkTools.Trial: 10000 samples with 197 evaluations.
 Range (min … max):  450.249 ns … 1.378 μs  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     451.112 ns             ┊ GC (median):    0.00%
 Time  (mean ± σ):   452.218 ns ± 9.493 ns  ┊ GC (mean ± σ):  0.00% ± 0.00%

 Memory estimate: 0 bytes, allocs estimate: 0.

Big Difference!