Short answer. No. Article 1210 states that the indivisibility of an obligation does not necessarily give rise to solidarity, and solidarity does not itself imply indivisibility either. The two concepts are independent; an obligation can be indivisible among several debtors without any of them being solidarily liable for the whole debt.
What the law says
The indivisibility of an obligation does not necessarily give rise to solidarity. Nor does solidarity of itself imply indivisibility.
Civil Code, Article 1210 — Indivisibility vs. Solidarity. Read the full provision →
Two different questions, kept separate
Article 1210 exists specifically to stop these two ideas from being confused with each other. It states plainly: The indivisibility of an obligation does not necessarily give rise to solidarity. Nor does solidarity of itself imply indivisibility. Indivisibility asks whether the object of the obligation, the thing to be given or the act to be done, can physically or legally be split into parts. Solidarity asks something else entirely: whether each debtor can be held liable by the creditor for the entire obligation, rather than only a proportionate share of it.
An indivisible obligation without solidary debtors
Because the two are independent, several debtors can jointly owe something that cannot be divided, such as delivering one specific horse or building one house, without any single one of them being answerable for the whole obligation on their own. In that situation, all the debtors ordinarily have to act together to comply, precisely because the thing itself cannot be split, but the creditor cannot simply pick one debtor and demand full performance from that person alone the way solidarity would allow.
Solidary debtors without an indivisible obligation
The reverse combination exists too. Debtors can be made solidarily liable, meaning the creditor may demand the entire debt from any one of them, over an obligation that is perfectly divisible, such as a sum of money that could easily be split into proportionate shares. Solidarity in that case comes from the law or from what the parties agreed, not from anything about whether the object of the obligation can be divided, which is exactly the point Article 1210 is making.
Why the distinction matters practically
Whether you can collect the full amount from any one debtor depends on whether the obligation is solidary, a question answered by the source of the obligation, the law, or what was agreed. Whether performance has to be delivered as a single, undivided whole depends on the nature of the thing owed. A creditor dealing with multiple debtors needs to check both questions separately, because assuming solidarity from indivisibility, or the other way around, can lead to demanding the wrong thing from the wrong person.