No, current SHA256 ASIC miners can't. Scrypt algorithm has loops which require a lot of memory and all operations have to be sequential which can't be made parallel, so it needs full computing environment (cpu-ram) for one hashing process. The algorithm was designed in this way to avoid hacking.
None of that is correct. SHA-256 miners can never run Scrypt just like they can never run SHA-512 They are designed to do one thing and one thing only.
However "LTC Scrypt" uses a mere 128KB of RAM. It all occurs on the GPU die (which has more than enough register space and L2 cache to hold the scratch pad). GPU memory latency to main memory (i.e. the 2GB of RAM on a graphics card) is incredibly long and the memory latency from GPU die off card to main memory is measured in fractional seconds. Utterly useless for Scrypt. If LTC required that to be used, a GPU would be far inferior to CPU with their 2MB+ of L2 and 6MB+ of L3 low latency cache. "Luckily" the modified parameters selected for LTC use a tiny fraction (~1%) of what is recommended by the Scrypt author for memory hardness even in low security applications and roughly 1/6000th of what is recommended for high security applications. It makes the scratchpad just small enough to fit inside a GPU and allow significant acceleration relative to a CPU.
Try bumping the parameters up just a little, GPU performance falls off a cliff while CPU performance is far more gradual. It doesn't matter if you attempt this on a system with 16GB (or even 32GB) of main memory. You can even try using a 1GB vs 2GB graphics card with negligible change in performance. The small memory scratchpad ensures neither a GPU main memory or the computer's main memory is used. The cache, inside the CPU die for CPU mining, or inside GPU die for GPU mining is what is used. Ever wonder why GPU accelerated password cracking programs don't include scrypt? The default paramters make the average GPU execution time <1 hash per second. Not a typo. Not 1 MH/s or 1 KH/s but <1 hash per second.
That is why "reaper" was so revolutionary but only for the weakened version of Scrypt used by LTC. It requires much less memory but still too much memory for a single SIMD unit and GPU main memory has far too much latency. That makes LTC impossible to mine on a GPU right? Well people thought so for a year. Reaper used a workaround by slaving multiple SIMD units together it stores the scratchpad across the cache and registers of multiple SIMD units. Now this reduces the parallelism of the GPU (which is why a GPU is only up to 10x better than a CPU vs 100x better on SHA-256). The combined register/cache across multiple SIMD units is large enough to contain the Scrypt scratchpad. This wouldn't be possible at the default parameters (~20MB of low latency memory) but it certainly possible at the reduce parameters used by LTC.