Short answer. Yes, so far as the rest is relevant. Where only part of a deposition is offered in evidence, the adverse party may require the offeror to introduce all of it which is relevant to the part introduced, and any party may introduce any other parts.
What the law says
(d) If only part of a deposition is offered in evidence by a party, the adverse party may require him or her to introduce all of it which is relevant to the part introduced, and any party may introduce any other parts.
Rule 23, Section 4 — Use of depositions. Read the full provision →
Two separate rights in one sentence
The paragraph gives you more than one option, and they work differently. First, as the adverse party you may require the offeror to introduce all of the deposition which is relevant to the part they introduced. Second, and independently, any party may introduce any other parts. The first puts the burden on the party who chose to open the transcript; the second lets you put in passages yourself.
The limit is relevance to what was offered
The compulsion is not unlimited. What you may require the offeror to introduce is all of the deposition which is relevant to the part introduced. So this is not a route to having the entire transcript read regardless of subject. If the excerpt concerned one meeting, the answer is the rest of what was said about that meeting — the qualification, the correction, the exchange that follows. The second limb is where genuinely separate passages come in.
Why the rule exists
A deposition is a continuous examination, and answers take their meaning from the questions around them. An admission followed immediately by an explanation is a different piece of evidence from the admission standing alone. The paragraph prevents a party from constructing a version of the testimony that the witness never gave by choosing where to start and stop, without requiring the court to sit through material nobody relies on.
Using it well
This works best prepared rather than improvised. Read the transcript before the hearing and mark, for each passage the other side is likely to use, the surrounding material that changes its sense — and separately, the passages you would offer in your own right. Remember that the general conditions still apply: a deposition may be used only so far as admissible under the rules of evidence, and against a party who was present, represented, or given due notice at its taking.
What the paragraph does not do
It does not make otherwise inadmissible material admissible. The passages you bring in are still subject to the rules of evidence, and to the general conditions governing every use of a deposition. Nor does it entitle you to have the whole transcript treated as evidence merely because a fragment was offered — the first limb is bounded by relevance to the part introduced. And it operates only once a party has actually offered part of a deposition; it is not a route to putting an unused deposition before the court.