By the time when Arrhenius proposed this theory he did not really have any argument supporting it.

Now, there exists only one argument supporting it.

The last common ancestor of all cellular living beings that exist on Earth was already a quite complex bacterium.

There is no doubt that it was the product of an already very long evolution process. For instance the genetic code that is used, with very small variations, by all living beings on Earth must have succeeded a long sequence of simpler genetic codes, with an increase of each step of the complexity of the code and of the number of amino-acids that could be encoded.

The oldest versions of the genetic code are likely to have encoded only between 4 and 6 amino-acids instead of 20 to 22, like today.

Based on the probable bacterial fossils that are quite old, it seems like the time from the apparition of life on Earth might have been too short to explain the complexity of the last common ancestor of the present living beings.

So this supports the idea that life could have appeared elsewhere, but then some bacteria and viruses have reached Earth and then they have evolved further.

Even in the unlikely case when this supposition were true, this changes nothing about the appearance of life, it just pushes it to another place that must have had a pretty much identical environment with the primitive Earth, in order to make possible the apparition of life.

Life cannot appear without a continuous source of energy for it. There exists only one known source of energy that can be used by the simplest possible forms of life, and this source of energy is the internal heat of a relatively big planet or of a very large satellite, like Titan or the big satellites of Jupiter.

The internal planetary heat can provide the energy for sustaining life indirectly, through volcans or hydrothermal vents. When volcanic rocks are ejected from the hotter inside of a planet, they consist of chemical substances that are no longer in chemical equilibrium at the lower temperature of the planet surface. This causes chemical reactions that result in substances like free dihydrogen, which, in the presence of catalysts, make possible the continuous synthesis of the complex organic molecules required for life.

As far as we know, Earth had ideal conditions for the appearance of life right here. It did not need to be colonized by bacterial spores coming for elsewhere.

The only reason why there is a very small chance for Arrhenius to have been right, is that the bigger Earth has remained very hot for a longer time than smaller planets like Mars, delaying the apparition of life here.

So it might have been possible for a place like Mars to have conditions suitable for the appearance of life before Earth. Life could have been appeared there and it could have been transported by one of the many meteorites that are known to have come from Mars to Earth as a consequence of big impacts.

Then Mars has lost most of its atmosphere and it became very cold, so if it ever had life, that could have disappeared.

For now this scenario that would match the theory of Arrhenius cannot be considered as 100% excluded, but in any case it is far-fetched and it does not change anything about the evolution of the living beings known on Earth, even if the initial part of that evolution could have taken place elsewhere, but in conditions not really different from those of the primitive Earth.