Is Earth inhospitable?
I think that your question is not really answerable.  But perhaps the reasons it's not answerable are themselves interesting.  There are two parts to this answer:


why do we think abiogenisis is likely?
is Earth, today, hospitable for organisms which originate through abiogenesis, if it is likely?


My answer to the first part is, I think, more well-founded than my answer to the second (I'm a physicist not a biologist!).

Is abiogenesis likely?

Well, we have exactly one example of a planet we can study closely with life on it (indeed there is only one example of a planet that we know has life at all at present), and life has been there for such a long time and has modified the environment of that planet so extensively that I think speculation about how life originated on that planet is, well, speculation.  That doesn't mean it's not interesting or that there might not be plausible and attractive hypotheses.

As an example it's tempting to argue that, because life originated on Earth almost as soon as it could, and disregarding panspermia, that abiogenesis must therefore be very common.  That argument is wrong.  There are a huge number of potentially-earthlike planets: this paper (arXiv version) claims that around 10% of stars may have a planet that is about the same size as Earth and gets around the same amount of sunlight as Earth.  There may be around $10^{24}$ stars in the observable universe: based on this paper (arXiv) there may be around $2\times 10^{12}$ galaxies and assuming each galaxy has $10^{12}$ stars we get $2\times 10^{23}$ stars.  So this means perhaps $2\times 10^{22}$ potentially-earthlike-planets.  All we know is that life originated on one of those planets, somehow (and abiogenesis is only one of at least two options), and it did so rather early.  What does that tell us?  Well, statisticians can play games and those games are probably reasonable, but it doesn't tell us, for sure, that the chance of that happening is higher than $1/10^{22}$.

This argument is related to the anthropic principle which I find rather unappealing in terms of explanatory power: I find the idea that the fine-structure constant must be what it is because otherwise we would not be here to measure it unappealing for instance.  But the anthropic principle is nevertheless not an idea that can be ruled out in general: we know life evolved on Earth because we're here to observe it, but this doesn't, yet, tell us anything about the probability of life evolving elsewhere: the universe could have just thrown a $10^{22}$-sided dice $10^{22}$ times and come up with the side that says 'life' once.

This doesn't mean that abiogenisis can't be really common, or even that it not likely that it is, it just means that we don't know that it is.  With luck this will change in the relatively near future, as we begin to be able to observe expolanets which convincingly have life.  Then we'll be able to use statistics in a more useful way.

How hospitable is Earth, today?

Well, let's assume abiogenisis is common, what can we say about Earth's hospitability to new life generated that way?  Well, we know that organisms on Earth are continually under attack by other organisms and things which may be organisms (I'm not a biologist, I'm not sure if viruses count as organisms or not).  And we know that these attacks can be pretty successful: for instance when Cortés invaded what is now Mexico the population was around 25-30 million, 50 years later it was around 3 million, and most of this decline happened due to disease (Wikipedia).

So life on Earth exists in this continual arms-race: things try to attack organisms and organisms develop hugely sophisticated immune systems to deal with those attacks.  We're living, right now, through an example of what happens when an agent attacks us for which our immune system is not yet prepared: fortunately it's not going to kill that many of us.

So let's propose that abiogenesis happens on Earth, now.  There are really two options.


Whatever appears is, say, RNA-based, and as such is recognisable by the existing warring factions.  It has no defences and gets promptly attacked and eaten.
Whatever appears is based on some significantly different chemistry.  But whatever it is, it's harvesting energy from somewhere and making it into itself.  As soon as it starts harvesting enough energy it becomes interesting to existing organisms who would be quite interested in getting that energy for themselves, who pretty quickly evolve to being able to do so and eat it.


The third option is that whatever it is turns out to be so efficient in some way that it wins the fight with the existing stuff.  I suspect, without any kind of proof, that this option is absurdly unlikely to happen.

I think what is more likely to happen is that, any time some new way of chemically stashing energy happens, it becomes attractive to existing organisms, who evolve to be able to harvest that energy and do so, long before whatever thing has happened could be described as 'alive'.

In summary: Earth is almost certainly not currently hospitable to newly-generated life, at all, because the existing life will eat it, one way or the other.