At scales of gigaparsecs, the universe appears to be both spatially homogeneous and isotropic, with matter denser at higher redshift. There is no way to generate this configuration of matter from a white hole. White holes are very much spatially anisotropic, that is you can point to where they are: left of you but not right of you, in front of you but not behind you, etc. The trajectories of the centres of momentum of galaxy clusters in every direction and at every redshift are diverging but not following an inertial radial. The really compelling accessible evidence against a white hole big bang is in the https://en.wikipedia.org/wiki/Lyman-alpha_forest from hydrogen clouds at different redshifts being backlit by bright background (and thus earlier in the universe) sources. This "stacking" is so hard to conceive for a white hole big bang that as far as I know, nobody has actually written down a reasonable explanation for it. More technically, (general relativity) metrics at those "cosmologically large" scales are so different from the metrics of white holes that we would already have noticed.
Additionally, a white hole big bang has to account for the chemistry of the earliest visible stars, whose spectral lines indicate essentially hydrogen with a tiny amount of helium and lithium. The standard cosmology's big bang nucleosynthesis's chemical abundances would have to be duplicated by a white hole big bang, and as far as I know nobody knows how to make that work. How does whatever is spit out of a white hole interior, or reflected off a white hole boundary, eventually turn into atomic hydrogen (with a smattering of heavier atoms up to 7 atomic mass units)? How does it account for the cosmic microwave background, which implies that in some region of the white hole big bang universe atoms practically everywhere were so hot as to be completely ionized?
Likewise, the interior metrics of black holes are very different from metrics which reasonably describe observations at gigaparsec scales. So if one takes a "wormhole" maximal extension of a black hole spacetime and throws enormous amounts matter into the black hole end, what emerges at the white hole end is nothing like a cosmology. Nor is what's inside. So a reasonable prediction of our galaxy's future is not exiting from a white hole (with us presently inside), and a reasonable prediction of our past is not that our galaxy was spit out of a white hole (with us presently outside).
As far as I know, nobody has done a convincing treatment of a non-eternal white hole in General Relativity. We've had theoretical models for forming black holes (i.e., there is a past of the black hole where there is no black hole yet, so it's non-eternal) since the 1930s (<https://en.wikipedia.org/wiki/Oppenheimer%E2%80%93Snyder_mod...>), and models for evaporating black holes since Hawking 1970, so it's quite striking there's nothing similar for white holes. There is ample astrophysical evidence supporting a range of models for non-eternal black holes and their horizons. There isn't anything seen by our telescopes that looks like it might be a white hole horizon.
The accelerated expansion of the universe, discovered in 1998, kills off a bunch of vaguely related ideas, like we are in a tiny tiny fraction of a thin shell (thin as in small compared to the shell's diameter) of matter that is expanding inertially from an initial impulse (an "explosion", figuratively) acting on a dense phase of matter. By "tiny tiny" here we're talking about our visible universe occupying less than 10^-23 of the shell's thickness. This is from a family of ideas tossed around a bit before Alan Guth published his 1997 book <https://en.wikipedia.org/wiki/The_Inflationary_Universe>. One could imagine a reflector onto which the shell fell, and that reflector could resemble a white hole; you can get a cyclic cosmology by having the entire shell eventually reconverge on the central reflector. However, we now see the expansion of our universe is speeding up rather than slowing down, so this doesn't work.
Finally, we're learning a lot more about the gravitational wave background of our universe. It is reasonable to bet that studies of the polarization of the cosmic microwave background (e.g. <https://lweb.cfa.harvard.edu/~cbischoff/bicepkeck/>) will find evidence of gravitational waves at scales large enough and uniform enough to rule out any sort of hole-like boundary (i.e., a one-way-crossing-only surface which has a spatial direction as well as being in the past) during the inflationary epoch. And we are probably already at the ruling-out point with baryon acoustic oscillation galaxy filament https://svs.gsfc.nasa.gov/13768 evidence.