No. Although the law gives employers a great deal of leeway in deciding whether to fire an employee, there are limits. State and federal laws prohibit employers from firing workers for certain reasons -- for example, because of the worker's race or religion, because the worker took family leave, or because the worker complained of illegal company activity. For a list of reasons for which you can never fire an employee, see Nolo's article Illegal Reasons for Firing Employees.
If the employee has an employment contract or you have made promises to the employee, then you must honor your commitment. Typically, this means you cannot fire the employee without "good cause."
For more on firing an employee with an employment contract, including an explanation of "good cause," see Nolo's article Firing Employees With Employment Contracts.