Yes, to satisfy the recertification requirement for your CCNP R&S certification, you can take and pass any one professional level exam on or before your certification expiration date.
Candidates are not required to take the Associate and Professional exams in any particular order, so you can take any exam from the CCNP Service Provider track without a valid pre-requisite. So yes, you are eligible to take exams from the CCNP Service Provider track. However, if you're looking to become certified in CCNP Service Provider, then you will need a valid pre-requisite as part of the necessary requirements shown here: Service Provider (CCNP SP)
To become certified in CCNP Service Provider, you will need a valid pre-requisite AND you will also need to pass the four required exams: 642-883 SPROUTE, 642-885 SPADVROUTE, 642-887 SPCORE, and 642-889 SPEDGE
Since each professional level exam is valid for 3 years, you would have 3 years to complete the requirements, starting from the date of your first passed exam for this certification.
I hope this helps and good luck!
Cisco Learning Network Moderator