Nice thing about the bell curve, it's an eigenfunction of the Fourier transform, down to proportional constants.
Okay, bigwords, let me explain.
"Eigenfunction" here just means it looks the same after FT. A Gaussian pulse has a Gaussian spectrum and vice versa.
So what? You can construct a filter with a Gaussian frequency response, and a short impulse through it will have a Gaussian waveform. (The pulse must be much shorter than the time constant of the filter, otherwise it smears out a bit -- somewhere between an impulse, and a square wave with sigmoidal rising and falling edges.)
Downside: the Gaussian function is not a rational function, i.e., it's composed of elementary functions, not a ratio of polynomials. The best you can do is a truncated polynomial best-fit. (There are plenty of polynomial functions with similar hilly shapes, 1/(1+x^2) for example, but none of them have the exponential cutoff of the Gaussian -- which has consequences in terms of statistics, analysis, etc. In particular, the hilly shape is only in the frequency domain. Rational polynomials are not eigenfunctions of the Fourier transform, so you won't get the same waveform as spectrum, only arbitrarily close.)
And a best-fit means you're probably going to need a lot of poles, i.e., a lot of L and C (or R, C and opamps). Which means, while it could still be adjustable... you might want to look at
why you want this, rather than to just go and do it.

Another way to understand this limitation, is to consider causality: a Gaussian wave has infinite extents, no beginning or end, only exponential cutoffs in either direction. An arbitrarily long filter, can have arbitrarily long delay, but it still must have zero output for t < 0, and some approximation of an exponential tail for 0 < t << t_peak. (The exponential decay after t_peak is quite normal, at least.)
So a better question is, why do you need such a particular waveform?
Also, you provide some units (L, R, V), but not enough to even determine whether your request is fundamentally
possible -- that is, the pulse width and height. Presumably the width is greater than 2ms and the height is less than 500mA...
Given your last requirement, it seems you don't want a Gaussian as such, but perhaps a piecewise linear velocity or acceleration function? Which does not have exponential cutoffs (beyond the settling time inherent in the system), and has, say, quadratic sections transitioning between regions. These can be constructed by multiply integrating square pulses, the heights of which determine the curvature and the durations of which determine the breakpoints (if the output amplitude is constant, these might be timed by comparator, so that the integrator gain doesn't need to be trimmed).
Tim