I can imagine two possible scenarios for Linux (the kernel): companies either choose to double down on it as a collaborative venture in order to distribute the cost of verification, or it is abandoned in favour of vendors who provide verified kernels at tremendous expense but in hopes of locking their competition out of the market.
As for the rest of the open source ecosystem that goes into Linux (the operating system), it would probably be abandoned.
All the internet backbone routers, endpoint routers and switches, hardware firewalls, VPN concentrators, the SSH daemons, SSL software, RSA keyfobs and the like, the content delivery networks and DNS ecosystem, SSL public trust system, the connectivity providers from ISP networks and national and international fibre connections to cellular and wifi networks, web browsers which billions of people use to interact with untrusted content, (datacenters, AT&T Long Lines building style classic phone system, the postal service, electricity subsystems, food and water supplies...), even staying in tech you've basically got to exploit something else before you get to whatever underlying OS there is and even if you get to it there's not necessarily a need to attack it.
NotPetya which took down Maersk and did $300Mn of damages was apparently spread (through their Windows AD) by compromised admin accounts which they were lax at managing[1] rather than kernel exploits. The SolarWinds Orion security flaws were blamed on weak passwords, not OS kernel exploits. And if getting inside, something like last month's SystemD/polkit exploit[2] shows that attacking the kernel isn't always necessary for privilege escalation.
Linux the kernel is important but it's the heart inside the ribcage, not the first or last line of defense, or the main thing to target.
The point is that if legislation were introduced that resulted in liability it would likely completely decapitate the FOSS ecosystem (among other things).
Why would it? If faced with the choice of taking liability for using Linux or rebuilding Amazon shop + AWS + Kindle + Echo on Windows I'd guess Amazon would do the former, wouldn't you?
In the short term? Perhaps. They might just develop their own proprietary OS - they already design custom CPUs!
I imagine it would depend heavily on how large the liability was. I expect individual components would begin being replaced with "certified" commercial alternatives. If not existing ones, definitely new ones. Remember that they make money by selling to customers who would also be subject to the same rules. Look at healthcare, aviation, and finance for concrete examples of the effects (both negative and positive) that red tape has on software and IT policies.
There's an entire FOSS ecosystem and the vast majority of it is composed of small-ish slow moving projects. The tech industry is also an entire ecosystem full of small and medium sized players. Even if behemoths such as mainline Linux and AWS somehow survived unchanged I would expect a much greater chilling effect on smaller players that couldn't afford to take on such risks. New companies and software projects would become very difficult to get off the ground (healthcare is a good example here). With few to no new entrants forward progress would slow to an absolute crawl.
All of this has downstream effects. Fewer consumer devices running Linux would mean even less hardware support. Security related liabilities would almost certainly mean more vendor locked hardware. Would companies like Purism remain viable (or even legal)? The steady stream of new FOSS users and contributors would almost certainly dwindle.
Depending on how such regulation was written, could open source contributors themselves become liable for a freely provided product?
As for the rest of the open source ecosystem that goes into Linux (the operating system), it would probably be abandoned.