You might be able to run chrome on an ASIC. But if you did, whatever it mines in a tab would not make use of the ASIC hardware.
Firstly, an ASIC is not a general purpose computer, it cannot just run whatever software you want on it. It does have a small general purpose computer for managing the ASIC hardware, but that is a low powered computer. This low powered computer has a CPU which you could compile a version of Chrome for and copy the binaries to it. Then you could start an X server to get a GUI over the network and view it on another computer.
However, this would not be useful to you at all. Chrome would not be able to use the ASIC hardware. It can only use the general purpose CPU, which, in this case, is super low powered. It would not mine any more coins than a regular computer, in fact, it will most definitely mine much much less, if any at all.