Can a cleric select an alignment domain that is not offered by his deity?

No.

There are two restrictions, and both apply:

1. At 1st level, a Cleric may only select its domains from those of its deity
2. A Cleric may only select alignment domains if they match its own alignment

If you have a mathematical mind, a Cleric may choose alignment domains from the intersection of the set of domains offered by its deity and its own alignment.

Let's see some examples!

Fharlanghn domains are Balance, Celerity, Luck, Portal, Protection, Travel, and Weather.

A Cleric of Fharlanghn may not select any alignment domain, because Fharlanghn has none (as suited to its True Neutral nature).

Tyr domains are Good, Knowledge, Law, Retribution and War.

A Lawful Good Cleric of Tyr may select both Good and Law (but need not to).

A Lawful Neutral Cleric of Tyr may select Law, but not Good.

A Neutral Good Cleric of Tyr may select Good, but not Law.

An hypothetical¹ True Neutral Cleric of Tyr may select neither Good nor Law.

¹ Hypothetical because clerics are supposed to be within one step of their deity's alignment, and True Neutral is two steps from Lawful Good.

Can a cleric select an alignment domain that is not offered by his deity?

No.

There are two restrictions, and both apply:

1. At 1st level, a Cleric may only select its domains from those of its deity
2. A Cleric may only select alignment domains if they match its own alignment

If you have a mathematical mind, a Cleric may choose alignment domains from the intersection of the set of domains offered by its deity and its own alignment.

Let's see some examples!

Fharlanghn domains are Balance, Celerity, Luck, Portal, Protection, Travel, and Weather.

A Cleric of Fharlanghn may not select any alignment domain, because Fharlanghn has none (as suited to its True Neutral nature).

Tyr domains are Good, Knowledge, Law, Retribution and War.

A Lawful Good Cleric of Tyr may select both Good and Law (but need not to).

A Lawful Neutral Cleric of Tyr may select Law, but not Good.

A Neutral Good Cleric of Tyr may select Good, but not Law.

An hypothetical¹ True Neutral Cleric of Tyr may select neither Good nor Law.

¹ Hypothetical because (1) Clerics are supposed to be within one step of their deity's alignment, and True Neutral is two steps from Lawful Good and (2) as noted by Hey I Can Chan, Clerics can only be True Neutral if their deity is True Neutral.