The linked gcc documentation does not talk of Partial Initialization it just talks of (Complete)Initialization or No Initialization.

What is partial Initialization?

The standards do not define Partial initialization of objects, either there is Complete initialization or No-initialization. Partial Initialization is a non-standard terminology which commonly refers a situation where you provide some initializers but not all i.e: Fewer initializers than the size of the array or the number of structure elements being initialized.

Example:

int array[10]={1,2};//Case 1:Partial Initialization

What is (Complete)Initialization or No Initialization?

Initialization means providing some initial value to the variable being created at the same time when it is being created. ie: in the same code statement.

Example:

int array[10]={0,1,2,3,4,5,6,7,8,9};//Case 2:Complete Initialization

int array[10];//Case 3:No Initialization

Following quoted paragraph describes the behavior for Case 3.

If you do not initialize a structure variable, the effect depends on whether it is has static storage (see Storage Class Specifiers) or not. If it is, members with integral types are initialized with 0 and pointer members are initialized to NULL; otherwise, the value of the structure‘s members is indeterminate.

The rules regarding Partial Initialization(Case 1) are well defined by the standard and these rules do not depend on the storage type of the variable being initialized.
AFAIK, All mainstream compilers have 100% compliance to these rules.

Can someone please tell me what the C and C++ standards say regarding partial automatic structure and automatic array initialization?

The C and C++ standards guarantee that even if an integer array is located on automatic storage and if there are fewer initializers in a brace-enclosed list then the uninitialized elements must be initialized to 0.

C99 Standard 6.7.8.21

If there are fewer initializers in a brace-enclosed list than there are elements or members of an aggregate, or fewer characters in a string literal used to initialize an array of known size than there are elements in the array, the remainder of the aggregate shall be initialized implicitly the same as objects that have static storage duration.

In C++ the rules are stated with a little difference.

C++03 Standard 8.5.1 Aggregates
Para 7:

If there are fewer initializers in the list than there are members in the aggregate, then each member not explicitly initialized shall be value-initialized (8.5). [Example:

struct S {int a;char* b;int c;};
 S ss ={1,"asdf"};

initializes ss.a with 1, ss.b with "asdf", and ss.c with the value of an expression of the form int(), that is,0. ]

While Value Initialization is defined in,
C++03 8.5 Initializers
Para 5:

To value-initialize an object of type T means:
— if T is a class type (clause 9) with a user-declared constructor (12.1), then the default constructor for T is called (and the initialization is ill-formed if T has no accessible default constructor);
— if T is a non-union class type without a user-declared constructor, then every non-static data member and base-class component of T is value-initialized;
— if T is an array type, then each element is value-initialized;
— otherwise, the object is zero-initialized