Then you'd have ownership even more tied up into a very small number of families. Big buisness (I'm talking things like GM and Boeing, not Facebook) requires a lot of capital. You can get that two ways- a lot of small investors (which is what stock is) or a few very big investors. If you hold them personally liable, you can't go route A. Nobody would be able to afford the risk (not to mention the extreme difficulty of finding out who owes how much based on when shares were owned). The end result is that much of advanced industry wouldn't exist, and we'd have even more of a plutocracy.
The real answer is holding those who run the company liable. The CxOs, the board of directors, the presidents and vice-presidents. Any time a company is fined, someone from that company should be writing a check or going to jail- the highest level person who either knew or should have known had they been doing their job competently. That way you can maintain the benefits of distributed financing and bring accountability back into it.