In D&D 3.5, No
The basilisk (Monster Manual 23-4) is not otherwise noted as not being immune to the petrifying gaze attacks of others of its kind, so a basilisk is immune to the petrifying gaze attacks of others of its kind as well as its own petrifying gaze. That is, the Monster Manual Glossary on gaze says, "A creature is immune to gaze attacks of others of its kind unless otherwise noted" (310), and the Dungeon Master's Guide Glossary adds, "A creature is immune to its own gaze attack" (294).1
In Pathfinder, also No
The description of the universal monster ability gaze attack mirrors that of the Monster Manual for Dungeons and Dragons 3.5. That game's basilisk is likewise not called out as specifically vulnerable to other basilisks' gazes.
Avoiding and getting a stony gaze (or fist)
Immunity to petrification is difficult to acquire in Dungeons and Dragons 3.5; in core that's usually a special quality of the angels (MM 10-13) and eladrins (MM 93-5), for example. Changing form so as to gain such a creature's traits is likely the easiest way to gain such an immunity. Immunity to petrification in Pathfinder is easily acquired for 8,000 gp from the on-the-nose-named amulet of proof against petrification.
The eyes of petrification (DMG 256) (98,000 gp; 0 lbs.) are a magic item in Dungeons and Dragons 3.5 yet not Pathfinder, but, by way of consolation, Pathfinder's feat Cockatrice Strike lets a warrior's attack turn foes into statues.
Neither magic item mentions basilisk eyes as necessary for their construction.
1 While the Monster Manual Glossary on gaze (MM 309-10) recommends a creature effectively turn off its gaze attack by (I kid you not, basilisk fans) wearing a veil, the Dungeon Master's Guide on gaze attacks says that a creature with a gaze attack can turn it off and on (294). However, the 3.5 revision paid the DMG Glossary only scant attention (for example, the spell resistance entry refers to the power word spells as spells of the school of conjuration rather than enchantment), and the Monster Manual is, after all, the game's primary source for monsters, so a DM may reasonably rule that the Monster Manual's correct in having a basilisk lack an on/off switch for its petrifying gaze.