Hmm, if I might perform some grammatical analysis of those script quotes:
SCRIPT: "We... No... Everyone worked to save the planet's future for
nothing... It's all because of you! You killed it!" - Lucca
Serge killed "it," but what is it referring to? The future, specifically the saved future. I think we can all agree that a "good" future can be killed in many ways. Lavos is one such manner, but others could exist (a nuclear apocalypse being quite similar to the ruined future from CT, for example).
Another key word is "because." The future was "killed" (rather melodramatic of Lucca but it establishes a poetic rather than strictly literal nature to her words) as a result of Serge. This could be a result of him breathing, squishing a bug he wasn't supposed to, or any number of things resulting from Serge living. We don't even know if Serge is the direct or indirect cause; merely, he is the ultimate cause. From this it may be that he is the actor or that which is acted upon.
Really the only hard facts we get from this is that the "saved" future has, in that dimension, been discarded, and that Serge is the ultimate cause of that (the exact nature needs subsequent quotes to determine).
SCRIPT: "How could you? How could you do such a thing...!? It's all
your fault that the world is going to end up like this! It's
all your fault, Serge!!! Murderer!!!" – Marle
Here we have a little more information. If Ghost-Marle is correct, Serge did an action (he did such a "thing") that resulted in the world ending up like the Dead Sea. It still doesn't mean that the world is the result of a Day of Lavos event, just that the world ends up ruined ("like this"). Again, we have a bit of poetic license here; Serge has "murdered" the future. Which actually brings up an interesting quote from "Lynx." If I am recalling correctly, he refers to Serge once as the "Assassin of Time."
So from these two quotes so far we know that the saved future has been discarded as a result of something Serge "did." The problem is, "did" from whose perspective? That is, did Serge do something in the past or future that caused the death of the future? If he already did it, then the game is entirely dependent on joining the two dimensions (so that what ever Serge "did" might be erased in the process). If he is to do it, well...
SCRIPT: "But the future that was supposed to have disappeared is about
to be restored here... The future destruction of our planet is
going to become a reality in this world once again..." – Marle
This seems to indicate it is that exact same future from CT that is about to be restored, but again that isn't necessitated. What is the future that was supposed to have disappeared? Just a Lavos-created apocalypse or a ruined future in general? If the latter, then a ruined future could be restored without the Day of Lavos.
Here we have an odd conundrum. The discarded future "is about to be restored" but the Dead Sea already exists. It would seem then that the discarded future isn't "about to be" restored but that it had already been restored. Verb tense is particular important here. This relates to the above quote too; seemingly the Ghosts are claiming that Serge is going to do something that has already happened.
On one hand, this identifies that action that Serge performs to doom the future as taking place in Serge's Future. On the other hand, this means that Serge is essentially "fated" to destroy the future; it is already assured, as the Dead Sea already exists. Unless, of course, the Dead Sea isn't the actual future of Home World but a possible future, or even a location that has an odd connection to the DBT and is siphoning a discarded future back into reality. The Ghost-Children are then like the Ghost of Christmas Present; "I see a vacant enertron in the poor chimney–corner, and a robot without an owner, carefully preserved. If these shadows remain unaltered, the future will die."
SCRIPT: "The vengeance of the future we killed is about to begin...
With Serge serving as the trigger..." - Crono
And again here we have better confirmation that the Dead Sea comes about because of something Serge
will do, yet it
already exists. This is almost diametrically opposed to Chrono Trigger, where the future doesn't change unless people make it change (and doesn't exist as changed until that change occurs). Curiously, this may well mean that at the end of Chrono Cross, we do not have an ideal future but a ruined future. I think we all would agree that the "vengeance of the future we killed" is referring to Lavos/the Time Devourer. Which means that Serge is somehow activating (or "triggering") that vengeance.
From these quotes, I am inclined to discard the possibility that the future is a result of Serge merely being absorbed by the Time Devourer. From these quotes it does not seem to be tenable that Serge merely existing or not (or merely being absorbed by something else) is sufficient. Serge has to
do something, he has to be the ultimate actor that causes the future's downfall; being absorbed isn't Serge doing anything. Being absorbed might allow the Time Devourer to destroy the future, but then that is the Time Devourer, not Serge, who is the acting agent.
So then, we might ask, in what way, after the Dead Sea events, do Serge's actions allow the Time Devourer to enact its vengeance?
In the game we see two very obvious events that might fit this description:
1) Serge’s actions directly allow the Dragon Gods to remerge (and the whole Dragon God had already been absorbed by the Time Devourer, possibly as the DG’s future from the Reptite Dimension would have been sent to the DBT after the Entity drew it to the Keystone Dimension). Serge then gave the Time Devourer an avatar in reality.
2) Serge’s actions directly link the DBT to the real world by using the Time Egg as Opassa beach; a gate, such as the one produced by the egg, should be able to be used in both directions.
However, other actions may also have resulted in the ruined future. But this seems to give us a time frame from which to work: the dimensions split in 1010AD, but the action that restored an apocalyptic future occured after Serge talked to the Ghost Children (so possibly even the evaporation of the Dead Sea is what results in a ruined future).