A opening bid of 1Club if followed by 1Diamond
For the time being we would assume you are playing a Green system similar to SAYC.

There are mainly 3 ways of responding to a natural / semi-natural (2+) non-forcing 1C opening:


Up-the-line: You bid your 5-card suits if you have one. Otherwise you simply bid your lowest 4-card suit. Under this method, a 1D opening does NOT deny holding a 4-card Major - you may have zero, one or two 4-card Majors. Say you hold: 963 KJ87 QJ53 A8 , you will usually respond 1D. Opener may either raise, bid his own 4-card Major, rebid Clubs or just bid 1NT. This is very natural and practically does not require extra gadgets. 
Walsh: You always bypass any length in Diamond in favour of a Major suit response, unless you hold game-forcing value. Therefore hands like 9 KJ87 QJT753 Q8 are still responded with a 1H, despite you holding much longer Diamond. This has the advantage of letting you find any 4-4 Major fits half a round earlier, but at the cost of bidding space: you are making a higher-level response (1M consumes more bidding space than 1D) which does not contain more information (Under this scheme, 1M response happens more often than 1D) than a lower response.
Transfer Walsh: In order to solve the dilemma between MAFIA (MAjor FIrst Always) and bidding space consumption, Transfer Walsh is designed. You bid 1D to show Hearts, 1H to show Spades and 1S to show Diamond or other minor-oriented hands. This keeps your bidding low while at the same time allowing you to prioritize showing Majors. However, this needs extra agreements with your partner about continuations, and your partnership must be able to cope with interference. 


Therefore in response to your question:

Q1: ... shows no [4-card] Major...

A1: Not necessarily. Given your question suggesting 1D denying Major, I assume you are playing Method 2. It usually denies 4-card Major, but only usually, not always. A scenario that you can still hold 4-card Major is that you have game-forcing values and your Diamond is worth mentioning, e.g. x AQxx KQJxxx Qx . You can respond 1D first, and upon opener rebidding 1NT you can reverse into 2H, or if he bids 1S you can make a XYZ or 4SF to further investigate into an ideal game or slam.

Q2: ... and 6-7 points...

A2: No, suit responses are unlimited. Your next call will most likely reveal what strength your hand has. A weak hand will pass, make a preference or rebid his own suit, a stronger hand will move on to game or make forcing bids to find out more about opener's hand.