The President does not have the power to demote or remove judges. In fact, no one can demote a judge. Congress can impeach and remove a judge, but it requires a two-thirds majority to do so. That is difficult to do for political reasons, as in the United States it is rare that either major political party holds less than a third of the Senate. So either party can block partisan actions by the other. An impeachment can only succeed with support from both major parties.
Looking specifically at current events, it seems unlikely that Republicans would go along with an attempt by Hillary Clinton to change the Chief Justice. And Democrats are unlikely to have more than fifty-four Senators (including Angus King and Bernie Sanders, who were elected as independents) after the current election. Most estimates are closer to an even split, possibly even a Republican advantage.
The other answer suggests a theoretical argument that the president doesn't have the power to appoint a Chief Justice. That may be a more practical way of doing this. However, there is a long custom of presidents appointing chief justices. While that's not a legal precedent, the court often pays deference to such customs--particularly one that goes back to a time when the writers of the constitution were alive and in Congress to object. In this case in particular, justices might be concerned that this would create a precedent for political interference with the court. That said, there's no legal precedent blocking it. It's possible.
In terms of the law, the relevant section would seem to be 28 Part I Chapter 1 Section 3 which just says
Whenever the Chief Justice is unable to perform the duties of his office or the office is vacant, his powers and duties shall devolve upon the associate justice next in precedence who is able to act, until such disability is removed or another Chief Justice is appointed and duly qualified.
It does not explain how a chief justice is appointed or qualified. Nor does that appear in the other sections of chapter 1.