Generally, yes, since the partial implementation of the Wright Committee's recommendations in 2010 under the coalition government, chairs of most select committees are elected by the whole House. A list of these committees can be found in Standing Order 122B.
However, several select committees, including the Intelligence and Security Committee, have a chair elected by their members. In this case, the Justice and Security Act 2013 sets out who is eligible to become a member of the committee in the first place, and the procedure which governs the election of the chair:
(4) A person is not eligible to become a member of the ISC unless the
- (a) is nominated for membership by the Prime Minister, and
is not a Minister of the Crown.
(5) Before deciding whether to
nominate a person for membership, the Prime Minister must consult the
Leader of the Opposition.
(6) A member of the ISC is to be the Chair
of the ISC chosen by its members.
That being said, there are ways in which the chair could be removed by No. 10, several of which are explained in Schedule 1 of the Act:
(2) A member of the ISC vacates office if—
- (a) the person ceases to be
a member of the House of Parliament by virtue of which the person is a
member of the ISC,
- (b) the person becomes a Minister of the Crown, or
- (c) a resolution for the person’s removal is passed in the House of
Parliament by virtue of which the person is a member of the ISC.
So Lewis could be removed from the committee by making him a government minister (highly unlikely), or by voting for his removal in the Commons (more likely). This would allow a new chair to be elected. As the government has a healthy Commons majority, this could be accomplished as soon as time could be allocated in the Commons timetable. It goes without saying that this would be unpopular politically, however given that Lewis has also lost the Conservative whip, we could be past that point.
Alternatively, No. 10 could pass primary legislation to reform the election of the committee chair, bypassing the above legislation. Perhaps the best way to do this, politically speaking, would be under the guise of implementing the rest of the Wright Reforms; in 2019, the Liason Committee recommended that election of select committee chairs by the whole House be extended to cover all committees:
The direct election of most select committee chairs by secret ballot
of the whole House has led to more confident committees, with an
increasing willingness to innovate and push the boundaries. We
recommend that the relevant changes to Standing Order No. 122B be made
to extend chair elections to all select committees.
Bear in mind, however, that primary legislation would take a while to pass through Parliament, and assuming that the Wright Committee recommendations are adhered to, the election would still be by secret ballot, meaning that the government could not exert pressure on its MPs to elect their preferred candidate.