I do not know if and how Intel could mitigate this in hardware, but if they can, Ice Lake will be a microarchitecture change so perhaps then.
Ice Lake has already "taped out" (IOW the design is complete and ready for testing and fabrication). No changes are possible. Tiger Lake is an optimization of Ice Lake with the same microarchitecture and is due in 2019. Maybe they could slightly tweak that (add a register or address mode) but not significant architectural changes. The problem is that fully mitigating Spectre might require significant architectural changes, not just tweaks. If so that is probably five years out or more.
Meltdown and Spectre are not Intel-only. Meltdown also affects some ARM CPUs and may affect IBM PowerPC. Spectre affects most CPUs designed since the mid-1990s.
Spectre (which is far more serious and difficult to mitigate than Meltdown) apparently exists for any CPU from any vendor which does speculative execution and branch prediction. That includes AMD Ryzen, ARM, IBM Power8, Power9, the System Z CPUs used in IBM mainframes, and possibly Oracle (formerly Sun) SPARC CPUs. It apparently affects a PowerMac G5 with a PowerPC 970FX CPU. It might include older CPUs such as the MIPS R10000, DEC Alpha 21264 and even the Motorola 68060, since I think they all use branch predictors and speculative execution.
Upgrading to a newer Intel or other brand CPU likely won't fix this, although specific hardware mitigations might lessen the impact or narrow the exposure. Research is ongoing, but the long term issue to be resolved is whether adding another layer of architectural patches on top of current CPUs will provide sufficient immunity to Spectre, or if all CPUs must be fundamentally redesigned and all software rewritten. E.g, migrating all computing to a hypothetical Itanium Mark II. The initial assessment from US-CERT, a US government cybersecurity group, said Spectre could not be reliably fixed in software or microcode and total replacement of all CPUs was the only true solution. They have since walked back that statement, but it shows how this is not a wild, unfounded possibility.
Meltdown is quick and easy to fix, albeit at a significant performance cost for some IO-intensive workloads. Intel "Haswell" CPUs and later already have PCID which limits the performance hit.
Spectre is far more serious. It allows a user process running within a VM to break out of this and access data in the hypervisor, which it can then trick into passing it data from other VMs. IOW on a big virtualized server running separate instances of SQL Server or Oracle each in their own VM, Spectre can cross VM boundaries and access in-memory user data from other VMs. On a Mac if you are running Parallels Desktop or VMWare Fusion, Spectre theoretically allows a Windows app to break out of that VM and access data in the host OS or from other Mac apps.
Meltdown is a single well-defined behavioral characteristic of Intel and certain ARM CPUs and maybe PowerPC. There currently seems little doubt when it's fixed in software via kernel page table isolation, that fix is reliable and total.
By contrast Spectre is more of a general method, of which two examples are currently known. It is much more difficult to mitigate and even when done, there is less confidence it's totally fixed. It's possible other Spectre variants will be discovered.
The performance cost of total mitigation can also be compared. Meltdown can be immediately and totally fixed by using OS kernel page table isolation (not patching apps) which incurs a variable performance cost from essentially zero to maybe 30% in extreme cases.
By contrast the only way to immediately achieve similar confidence in a Spectre fix would be to disable branch prediction and instruction speculation. That would probably have a 5x or more performance hit -- the CPU would essentially be unusable. Thus Spectre fixes to date have been more like patching holes in a leaking dike. Over the past few days there is increasing confidence maybe Spectre can be adequately mitigated by various software and microcode patches but it's not totally certain. It might not be totally fixable without a totally new CPU architecture, which would imply all software would have to be recompiled or re-written to run on that new architecture. However the Apple Mac has changed CPUs twice and it's obviously still here and going strong, so it's theoretically possible on a broader scale.