Generally no. The Bill of Rights provides protection for individual liberty from actions by government officials. This is called the state-action doctrine. Private property is not government-owned. Restrictions on individuals’ free-speech rights on private property do not involve state action.
However, a few states have interpreted their own state constitutions to provide even greater free-speech protection than the federal Constitution offers. For example, the New Jersey Supreme Court has ruled that individuals have free-speech rights at privately owned shopping malls. Most state supreme courts that have examined the issue have disagreed. In April 2002, the Iowa Supreme Court refused to extend its definition of public property to include large, privately owned shopping malls.