Thanks for the question Aditya.
Review the Texas Property Code
You may want to look at the Texas property code, especially sections 92.001 (legal definitions for Residential Rental Properties), 92.103 (Obligation to Refund) and 92.1031 (Retention of Security Deposit). The text may be found here, and specifies under what conditions the landlord is obligated to refund the security deposit.
While I am not a lawyer, make no claim to have knowledge of Texas law, and do not intend to provide legal advice, the following may be generally helpful based on my professional experience in negotiating residential leases in my former state of residence:
The Contractual Perspective
Generally speaking, the security deposit is tied to a lease, which indicates a specific lease term and tenants. When the lease ends and the property is surrendered, the security deposit is payable to the named tenant(s). When a lease is amended or extended however, and the property is not surrendered, the landlord will usually retain the security deposit.
The landlord's obligations depend on a number of circumstances. When you say that you "continued" the lease, and that you are on the only tenant on the "new" lease, can you please confirm whether:
1) the original lease (where you were both tenants) ended, after which you entered into a new lease on your own, or
2) the original lease was amended/changed to remove the other tenant and/or change the length or other terms of the lease, or
3) the lease was not formally changed and the other tenant left based on a verbal agreement between you and/or the landlord?
A lease is a contract between the tenants and the landlord. In case 1) above, if the lease truly ended/expired, and you signed a new lease (in which case both of you would likely have been required to surrender the original keys and signed a new lease), it is likely that the landlord should have provided the inspection and refunded any remaining portion of the deposit to both of you. In this case, he would need to take up his concern with the landlord, not you, but my interpretation of how you phrased your question is that this is not what happened.
In cases 2) and 3), from the landlord's point of view, the original lease that you both signed has not expired, and you have not surrendered the property, so it's likely that the landlord has no legal obligation to refund your security deposit yet. If the lease was amended, it's likely that the other tenant would have had to sign this amendment, which may have changed the terms and length of the lease and removed him as a tenant. If you are now the only tenant on the lease, it is very possible that neither you or the landlord have any further obligation to him regarding the securing deposit. If he expected to receive a portion of his security deposit back, this is something that should have been worked out between the two of you before he left. It is likely that when your lease ends, the landlord will simply pay out the deposit (net of deductions) to you as the named tenant on the lease.
The Ethical Perspective
From an ethical perspective however, you may feel that this was an oversight, and that if you both paid in to the security deposit, that you should both share what you may get back at the end of the lease. If so, you can always provide him with half of what you get back when the lease ends and you surrender the property back to the landlord.
However, I don't see any justifiable reason for him to simply ask for half of the initial deposit back. This is not fair because it is asking you to take on an additional risk that you did not originally agree to take on. Originally, you both agreed to share the risk of damages for the full term of the lease. Then, he wanted to change the deal and get out of the lease early. Given that the landlord wasn't willing to inspect and assess at that point, he should have been aware of the consequences of his decision to leave. Asking you to take over 100% of the risk of damages is unfair, and you are not cheating him if you decide to not pay him until the lease ends and the refund is final. This would be the case even if you were not aware of any damage to the property. In the worst case scenario, if the landlord keeps 100% of the deposit, it would not be fair for you to have paid for all of the damages.
Your roommate may have been unaware that his part of the security deposit would remain at risk until the end of the lease, and this will be a lesson learned for him. I don't think you have an obligation to pay him before the landlord provides a final refund.
Again, please be aware that these laws and their application vary by state.
Best of luck in your decision.